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Lemniscate elliptic functions
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<h2 id="lemniscate-sine-and-cosine-functions">Lemniscate sine and cosine functions</h2> <h3>Definitions</h3> <p>The lemniscate functions sl and cl can be defined as the solution to the <a href="/facts/Initial_value_problem/PqmJfGML">initial value problem</a>:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> d d z sl z = ( 1 + sl 2 z ) cl z , d d z cl z = − ( 1 + cl 2 z ) sl z , sl 0 = 0 , cl 0 = 1 , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {sl} z={\bigl (}1+\operatorname {sl} ^{2}z{\bigr )}\operatorname {cl} z,\ {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cl} z=-{\bigl (}1+\operatorname {cl} ^{2}z{\bigr )}\operatorname {sl} z,\ \operatorname {sl} 0=0,\ \operatorname {cl} 0=1,} <p>or equivalently as the <a href="/facts/Inverse_function/Tg5nnXlu">inverses</a> of an <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integral</a>, the <a href="/facts/Schwarz%E2%80%93Christoffel_mapping/vzHbRDDG">Schwarz–Christoffel map</a> from the complex <a href="/facts/Unit_disk/phrUu7yC">unit disk</a> to a square with corners { 1 2 ϖ , 1 2 ϖ i , − 1 2 ϖ , − 1 2 ϖ i } : {\displaystyle {\big \{}{\tfrac {1}{2}}\varpi ,{\tfrac {1}{2}}\varpi i,-{\tfrac {1}{2}}\varpi ,-{\tfrac {1}{2}}\varpi i{\big \}}\colon } <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> z = ∫ 0 sl z d t 1 − t 4 = ∫ cl z 1 d t 1 − t 4 . {\displaystyle z=\int _{0}^{\operatorname {sl} z}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}=\int _{\operatorname {cl} z}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}.} <p>Beyond that square, the functions can be <a href="/facts/Analytic_continuation/aBCuEgvt">analytically continued</a> to the whole <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> by a series of <a href="/facts/Schwarz_reflection_principle/gsxRVe13">reflections</a>. </p><p>By comparison, the circular sine and cosine can be defined as the solution to the initial value problem: </p> d d z sin z = cos z , d d z cos z = − sin z , sin 0 = 0 , cos 0 = 1 , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\sin z=\cos z,\ {\frac {\mathrm {d} }{\mathrm {d} z}}\cos z=-\sin z,\ \sin 0=0,\ \cos 0=1,} <p>or as inverses of a map from the <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a> to a half-infinite strip with real part between − 1 2 π , 1 2 π {\displaystyle -{\tfrac {1}{2}}\pi ,{\tfrac {1}{2}}\pi } and positive imaginary part: </p> z = ∫ 0 sin z d t 1 − t 2 = ∫ cos z 1 d t 1 − t 2 . {\displaystyle z=\int _{0}^{\sin z}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}=\int _{\cos z}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}.} <h3>Relation to the lemniscate constant</h3> <p class="note">Main article: <a href="/facts/Lemniscate_constant/jzkcBdBL">Lemniscate constant</a></p> <p>The lemniscate functions have minimal real period 2<i>ϖ</i>, minimal <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary</a> period 2<i>ϖ</i>i and fundamental complex periods ( 1 + i ) ϖ {\displaystyle (1+i)\varpi } and ( 1 − i ) ϖ {\displaystyle (1-i)\varpi } for a constant <i>ϖ</i> called the <i><a href="/facts/Lemniscate_constant/jzkcBdBL">lemniscate constant</a></i>,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> ϖ = 2 ∫ 0 1 d t 1 − t 4 = 2.62205 … {\displaystyle \varpi =2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}=2.62205\ldots } <p>The lemniscate functions satisfy the basic relation cl z = sl ( 1 2 ϖ − z ) , {\displaystyle \operatorname {cl} z={\operatorname {sl} }{\bigl (}{\tfrac {1}{2}}\varpi -z{\bigr )},} analogous to the relation cos z = sin ( 1 2 π − z ) . {\displaystyle \cos z={\sin }{\bigl (}{\tfrac {1}{2}}\pi -z{\bigr )}.} </p><p>The lemniscate constant <i>ϖ</i> is a close analog of the <a href="/facts/Pi/vC0UBj1q">circle constant <i>π</i></a>, and many identities involving <i>π</i> have analogues involving <i>ϖ</i>, as identities involving the <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a> have analogues involving the lemniscate functions. For example, <a href="/facts/Vi%C3%A8te%27s_formula/EWC3rdTn">Viète's formula</a> for <i>π</i> can be written: </p><p> 2 π = 1 2 ⋅ 1 2 + 1 2 1 2 ⋅ 1 2 + 1 2 1 2 + 1 2 1 2 ⋯ {\displaystyle {\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots } </p><p>An analogous formula for <i>ϖ</i> is:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p><p> 2 ϖ = 1 2 ⋅ 1 2 + 1 2 / 1 2 ⋅ 1 2 + 1 2 / 1 2 + 1 2 / 1 2 ⋯ {\displaystyle {\frac {2}{\varpi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\Bigg /}\!{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\bigg /}\!{\sqrt {\frac {1}{2}}}}}}}\cdots } </p><p>The <a href="/facts/Machin-like_formula/tR2CvUWp">Machin formula</a> for <i>π</i> is 1 4 π = 4 arctan 1 5 − arctan 1 239 , {\textstyle {\tfrac {1}{4}}\pi =4\arctan {\tfrac {1}{5}}-\arctan {\tfrac {1}{239}},} and several similar formulas for <i>π</i> can be developed using trigonometric angle sum identities, e.g. Euler's formula 1 4 π = arctan 1 2 + arctan 1 3 {\textstyle {\tfrac {1}{4}}\pi =\arctan {\tfrac {1}{2}}+\arctan {\tfrac {1}{3}}} . Analogous formulas can be developed for <i>ϖ</i>, including the following found by Gauss: 1 2 ϖ = 2 arcsl 1 2 + arcsl 7 23 . {\displaystyle {\tfrac {1}{2}}\varpi =2\operatorname {arcsl} {\tfrac {1}{2}}+\operatorname {arcsl} {\tfrac {7}{23}}.} <a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>The lemniscate and circle constants were found by Gauss to be related to each-other by the <a href="/facts/Arithmetic-geometric_mean/8PRN9UB4">arithmetic-geometric mean</a> <i>M</i>:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p> π ϖ = M ( 1 , 2 ) {\displaystyle {\frac {\pi }{\varpi }}=M{\left(1,{\sqrt {2}}\!~\right)}} </p> <h2 id="argument-identities">Argument identities</h2> <h3>Zeros, poles and symmetries</h3> <p>The lemniscate functions cl and sl are <a href="/facts/Even_and_odd_functions/tGHWJqAa">even and odd functions</a>, respectively, </p> cl ( − z ) = cl z sl ( − z ) = − sl z {\displaystyle {\begin{aligned}\operatorname {cl} (-z)&=\operatorname {cl} z\\[6mu]\operatorname {sl} (-z)&=-\operatorname {sl} z\end{aligned}}} <p>At translations of 1 2 ϖ , {\displaystyle {\tfrac {1}{2}}\varpi ,} cl and sl are exchanged, and at translations of 1 2 i ϖ {\displaystyle {\tfrac {1}{2}}i\varpi } they are additionally rotated and <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocated</a>:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> cl ( z ± 1 2 ϖ ) = ∓ sl z , cl ( z ± 1 2 i ϖ ) = ∓ i sl z sl ( z ± 1 2 ϖ ) = ± cl z , sl ( z ± 1 2 i ϖ ) = ± i cl z {\displaystyle {\begin{aligned}{\operatorname {cl} }{\bigl (}z\pm {\tfrac {1}{2}}\varpi {\bigr )}&=\mp \operatorname {sl} z,&{\operatorname {cl} }{\bigl (}z\pm {\tfrac {1}{2}}i\varpi {\bigr )}&={\frac {\mp i}{\operatorname {sl} z}}\\[6mu]{\operatorname {sl} }{\bigl (}z\pm {\tfrac {1}{2}}\varpi {\bigr )}&=\pm \operatorname {cl} z,&{\operatorname {sl} }{\bigl (}z\pm {\tfrac {1}{2}}i\varpi {\bigr )}&={\frac {\pm i}{\operatorname {cl} z}}\end{aligned}}} <p>Doubling these to translations by a <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a>-Gaussian-integer multiple of ϖ {\displaystyle \varpi } (that is, ± ϖ {\displaystyle \pm \varpi } or ± i ϖ {\displaystyle \pm i\varpi } ), negates each function, an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a>: </p> cl ( z + ϖ ) = cl ( z + i ϖ ) = − cl z sl ( z + ϖ ) = sl ( z + i ϖ ) = − sl z {\displaystyle {\begin{aligned}\operatorname {cl} (z+\varpi )&=\operatorname {cl} (z+i\varpi )=-\operatorname {cl} z\\[4mu]\operatorname {sl} (z+\varpi )&=\operatorname {sl} (z+i\varpi )=-\operatorname {sl} z\end{aligned}}} <p>As a result, both functions are invariant under translation by an <a href="/facts/Gaussian_integer/zj82AhtL">even-Gaussian-integer</a> multiple of ϖ {\displaystyle \varpi } .<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> That is, a displacement ( a + b i ) ϖ , {\displaystyle (a+bi)\varpi ,} with a + b = 2 k {\displaystyle a+b=2k} for integers <i>a</i>, <i>b</i>, and <i>k</i>. </p> cl ( z + ( 1 + i ) ϖ ) = cl ( z + ( 1 − i ) ϖ ) = cl z sl ( z + ( 1 + i ) ϖ ) = sl ( z + ( 1 − i ) ϖ ) = sl z {\displaystyle {\begin{aligned}{\operatorname {cl} }{\bigl (}z+(1+i)\varpi {\bigr )}&={\operatorname {cl} }{\bigl (}z+(1-i)\varpi {\bigr )}=\operatorname {cl} z\\[4mu]{\operatorname {sl} }{\bigl (}z+(1+i)\varpi {\bigr )}&={\operatorname {sl} }{\bigl (}z+(1-i)\varpi {\bigr )}=\operatorname {sl} z\end{aligned}}} <p>This makes them <a href="/facts/Elliptic_functions/mF03oI04">elliptic functions</a> (doubly periodic <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic functions</a> in the complex plane) with a <a href="/facts/Square_lattice/mcVtpcQK">diagonal square</a> <a href="/facts/Period_lattice/DsPofQqG">period lattice</a> of fundamental periods ( 1 + i ) ϖ {\displaystyle (1+i)\varpi } and ( 1 − i ) ϖ {\displaystyle (1-i)\varpi } .<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Elliptic functions with a square period lattice are more symmetrical than arbitrary elliptic functions, following the symmetries of the square. </p><p>Reflections and quarter-turn rotations of lemniscate function arguments have simple expressions: </p> cl z ¯ = cl z ¯ sl z ¯ = sl z ¯ cl i z = 1 cl z sl i z = i sl z {\displaystyle {\begin{aligned}\operatorname {cl} {\bar {z}}&={\overline {\operatorname {cl} z}}\\[6mu]\operatorname {sl} {\bar {z}}&={\overline {\operatorname {sl} z}}\\[4mu]\operatorname {cl} iz&={\frac {1}{\operatorname {cl} z}}\\[6mu]\operatorname {sl} iz&=i\operatorname {sl} z\end{aligned}}} <p>The sl function has simple <a href="/facts/Zeros_and_poles/gnA3U8VP">zeros</a> at Gaussian integer multiples of <i>ϖ</i>, complex numbers of the form a ϖ + b ϖ i {\displaystyle a\varpi +b\varpi i} for integers <i>a</i> and <i>b</i>. It has simple <a href="/facts/Zeros_and_poles/gnA3U8VP">poles</a> at Gaussian <a href="/facts/Half-integer/oJ29mf8G">half-integer</a> multiples of <i>ϖ</i>, complex numbers of the form ( a + 1 2 ) ϖ + ( b + 1 2 ) ϖ i {\displaystyle {\bigl (}a+{\tfrac {1}{2}}{\bigr )}\varpi +{\bigl (}b+{\tfrac {1}{2}}{\bigr )}\varpi i} , with <a href="/facts/Residue_(complex_analysis)/GllQJk32">residues</a> ( − 1 ) a − b + 1 i {\displaystyle (-1)^{a-b+1}i} . The cl function is reflected and offset from the sl function, cl z = sl ( 1 2 ϖ − z ) {\displaystyle \operatorname {cl} z={\operatorname {sl} }{\bigl (}{\tfrac {1}{2}}\varpi -z{\bigr )}} . It has zeros for arguments ( a + 1 2 ) ϖ + b ϖ i {\displaystyle {\bigl (}a+{\tfrac {1}{2}}{\bigr )}\varpi +b\varpi i} and poles for arguments a ϖ + ( b + 1 2 ) ϖ i , {\displaystyle a\varpi +{\bigl (}b+{\tfrac {1}{2}}{\bigr )}\varpi i,} with residues ( − 1 ) a − b i . {\displaystyle (-1)^{a-b}i.} </p><p>Also </p> sl z = sl w ↔ z = ( − 1 ) m + n w + ( m + n i ) ϖ {\displaystyle \operatorname {sl} z=\operatorname {sl} w\leftrightarrow z=(-1)^{m+n}w+(m+ni)\varpi } <p>for some m , n ∈ Z {\displaystyle m,n\in \mathbb {Z} } and </p> sl ( ( 1 ± i ) z ) = ( 1 ± i ) sl z sl ′ z . {\displaystyle \operatorname {sl} ((1\pm i)z)=(1\pm i){\frac {\operatorname {sl} z}{\operatorname {sl} 'z}}.} <p>The last formula is a special case of <a href="/facts/Complex_multiplication/YQNcbNXP">complex multiplication</a>. Analogous formulas can be given for sl ( ( n + m i ) z ) {\displaystyle \operatorname {sl} ((n+mi)z)} where n + m i {\displaystyle n+mi} is any Gaussian integer – the function sl {\displaystyle \operatorname {sl} } has complex multiplication by Z [ i ] {\displaystyle \mathbb {Z} [i]} .<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>There are also infinite series reflecting the distribution of the zeros and poles of sl:<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> 1 sl z = ∑ ( n , k ) ∈ Z 2 ( − 1 ) n + k z + n ϖ + k ϖ i {\displaystyle {\frac {1}{\operatorname {sl} z}}=\sum _{(n,k)\in \mathbb {Z} ^{2}}{\frac {(-1)^{n+k}}{z+n\varpi +k\varpi i}}} sl z = − i ∑ ( n , k ) ∈ Z 2 ( − 1 ) n + k z + ( n + 1 / 2 ) ϖ + ( k + 1 / 2 ) ϖ i . {\displaystyle \operatorname {sl} z=-i\sum _{(n,k)\in \mathbb {Z} ^{2}}{\frac {(-1)^{n+k}}{z+(n+1/2)\varpi +(k+1/2)\varpi i}}.} <h3>Pythagorean-like identity</h3> <p>The lemniscate functions satisfy a <a href="/facts/Pythagorean_trigonometric_identity/MQsE3Qhc">Pythagorean</a>-like identity: </p> c l 2 z + s l 2 z + c l 2 z s l 2 z = 1 {\displaystyle \operatorname {cl^{2}} z+\operatorname {sl^{2}} z+\operatorname {cl^{2}} z\,\operatorname {sl^{2}} z=1} <p>As a result, the parametric equation ( x , y ) = ( cl t , sl t ) {\displaystyle (x,y)=(\operatorname {cl} t,\operatorname {sl} t)} parametrizes the <a href="/facts/Quartic_plane_curve/5j35LcSS">quartic curve</a> x 2 + y 2 + x 2 y 2 = 1. {\displaystyle x^{2}+y^{2}+x^{2}y^{2}=1.} </p><p>This identity can alternately be rewritten:<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> ( 1 + c l 2 z ) ( 1 + s l 2 z ) = 2 {\displaystyle {\bigl (}1+\operatorname {cl^{2}} z{\bigr )}{\bigl (}1+\operatorname {sl^{2}} z{\bigr )}=2} c l 2 z = 1 − s l 2 z 1 + s l 2 z , s l 2 z = 1 − c l 2 z 1 + c l 2 z {\displaystyle \operatorname {cl^{2}} z={\frac {1-\operatorname {sl^{2}} z}{1+\operatorname {sl^{2}} z}},\quad \operatorname {sl^{2}} z={\frac {1-\operatorname {cl^{2}} z}{1+\operatorname {cl^{2}} z}}} <p>Defining a <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">tangent-sum</a> operator as a ⊕ b := tan ( arctan a + arctan b ) = a + b 1 − a b , {\displaystyle a\oplus b\mathrel {:=} \tan(\arctan a+\arctan b)={\frac {a+b}{1-ab}},} gives: </p> c l 2 z ⊕ s l 2 z = 1. {\displaystyle \operatorname {cl^{2}} z\oplus \operatorname {sl^{2}} z=1.} <p>The functions cl ~ {\displaystyle {\tilde {\operatorname {cl} }}} and sl ~ {\displaystyle {\tilde {\operatorname {sl} }}} satisfy another Pythagorean-like identity: </p> ( ∫ 0 x cl ~ t d t ) 2 + ( 1 − ∫ 0 x sl ~ t d t ) 2 = 1. {\displaystyle \left(\int _{0}^{x}{\tilde {\operatorname {cl} }}\,t\,\mathrm {d} t\right)^{2}+\left(1-\int _{0}^{x}{\tilde {\operatorname {sl} }}\,t\,\mathrm {d} t\right)^{2}=1.} <h3>Derivatives and integrals</h3> <p>The derivatives are as follows: </p> d d z cl z = c l ′ z = − ( 1 + c l 2 z ) sl z = − 2 sl z sl 2 z + 1 c l ′ 2 z = 1 − c l 4 z d d z sl z = s l ′ z = ( 1 + s l 2 z ) cl z = 2 cl z cl 2 z + 1 s l ′ 2 z = 1 − s l 4 z {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cl} z=\operatorname {cl'} z&=-{\bigl (}1+\operatorname {cl^{2}} z{\bigr )}\operatorname {sl} z=-{\frac {2\operatorname {sl} z}{\operatorname {sl} ^{2}z+1}}\\\operatorname {cl'^{2}} z&=1-\operatorname {cl^{4}} z\\[5mu]{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {sl} z=\operatorname {sl'} z&={\bigl (}1+\operatorname {sl^{2}} z{\bigr )}\operatorname {cl} z={\frac {2\operatorname {cl} z}{\operatorname {cl} ^{2}z+1}}\\\operatorname {sl'^{2}} z&=1-\operatorname {sl^{4}} z\end{aligned}}} d d z cl ~ z = − 2 sl ~ z cl z − sl ~ z cl z d d z sl ~ z = 2 cl ~ z cl z − cl ~ z cl z {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\,{\tilde {\operatorname {cl} }}\,z&=-2\,{\tilde {\operatorname {sl} }}\,z\,\operatorname {cl} z-{\frac {{\tilde {\operatorname {sl} }}\,z}{\operatorname {cl} z}}\\{\frac {\mathrm {d} }{\mathrm {d} z}}\,{\tilde {\operatorname {sl} }}\,z&=2\,{\tilde {\operatorname {cl} }}\,z\,\operatorname {cl} z-{\frac {{\tilde {\operatorname {cl} }}\,z}{\operatorname {cl} z}}\end{aligned}}} <p>The second derivatives of lemniscate sine and lemniscate cosine are their negative duplicated cubes: </p> d 2 d z 2 cl z = − 2 c l 3 z {\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}\operatorname {cl} z=-2\operatorname {cl^{3}} z} d 2 d z 2 sl z = − 2 s l 3 z {\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} z^{2}}}\operatorname {sl} z=-2\operatorname {sl^{3}} z} <p>The lemniscate functions can be integrated using the inverse tangent function: </p> ∫ cl z d z = arctan sl z + C ∫ sl z d z = − arctan cl z + C ∫ cl ~ z d z = sl ~ z cl z + C ∫ sl ~ z d z = − cl ~ z cl z + C {\displaystyle {\begin{aligned}\int \operatorname {cl} z\mathop {\mathrm {d} z} &=\arctan \operatorname {sl} z+C\\\int \operatorname {sl} z\mathop {\mathrm {d} z} &=-\arctan \operatorname {cl} z+C\\\int {\tilde {\operatorname {cl} }}\,z\,\mathrm {d} z&={\frac {{\tilde {\operatorname {sl} }}\,z}{\operatorname {cl} z}}+C\\\int {\tilde {\operatorname {sl} }}\,z\,\mathrm {d} z&=-{\frac {{\tilde {\operatorname {cl} }}\,z}{\operatorname {cl} z}}+C\end{aligned}}} <h3>Argument sum and multiple identities</h3> <p>Like the trigonometric functions, the lemniscate functions satisfy argument sum and difference identities. The original identity used by Fagnano for bisection of the lemniscate was:<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> sl ( u + v ) = sl u s l ′ v + sl v s l ′ u 1 + s l 2 u s l 2 v {\displaystyle \operatorname {sl} (u+v)={\frac {\operatorname {sl} u\,\operatorname {sl'} v+\operatorname {sl} v\,\operatorname {sl'} u}{1+\operatorname {sl^{2}} u\,\operatorname {sl^{2}} v}}} <p>The derivative and Pythagorean-like identities can be used to rework the identity used by Fagano in terms of sl and cl. Defining a <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">tangent-sum</a> operator a ⊕ b := tan ( arctan a + arctan b ) {\displaystyle a\oplus b\mathrel {:=} \tan(\arctan a+\arctan b)} and tangent-difference operator a ⊖ b := a ⊕ ( − b ) , {\displaystyle a\ominus b\mathrel {:=} a\oplus (-b),} the argument sum and difference identities can be expressed as:<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> cl ( u + v ) = cl u cl v ⊖ sl u sl v = cl u cl v − sl u sl v 1 + sl u cl u sl v cl v cl ( u − v ) = cl u cl v ⊕ sl u sl v sl ( u + v ) = sl u cl v ⊕ cl u sl v = sl u cl v + cl u sl v 1 − sl u cl u sl v cl v sl ( u − v ) = sl u cl v ⊖ cl u sl v {\displaystyle {\begin{aligned}\operatorname {cl} (u+v)&=\operatorname {cl} u\,\operatorname {cl} v\ominus \operatorname {sl} u\,\operatorname {sl} v={\frac {\operatorname {cl} u\,\operatorname {cl} v-\operatorname {sl} u\,\operatorname {sl} v}{1+\operatorname {sl} u\,\operatorname {cl} u\,\operatorname {sl} v\,\operatorname {cl} v}}\\[2mu]\operatorname {cl} (u-v)&=\operatorname {cl} u\,\operatorname {cl} v\oplus \operatorname {sl} u\,\operatorname {sl} v\\[2mu]\operatorname {sl} (u+v)&=\operatorname {sl} u\,\operatorname {cl} v\oplus \operatorname {cl} u\,\operatorname {sl} v={\frac {\operatorname {sl} u\,\operatorname {cl} v+\operatorname {cl} u\,\operatorname {sl} v}{1-\operatorname {sl} u\,\operatorname {cl} u\,\operatorname {sl} v\,\operatorname {cl} v}}\\[2mu]\operatorname {sl} (u-v)&=\operatorname {sl} u\,\operatorname {cl} v\ominus \operatorname {cl} u\,\operatorname {sl} v\end{aligned}}} <p>These resemble their <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">trigonometric analogs</a>: </p> cos ( u ± v ) = cos u cos v ∓ sin u sin v sin ( u ± v ) = sin u cos v ± cos u sin v {\displaystyle {\begin{aligned}\cos(u\pm v)&=\cos u\,\cos v\mp \sin u\,\sin v\\[6mu]\sin(u\pm v)&=\sin u\,\cos v\pm \cos u\,\sin v\end{aligned}}} <p>In particular, to compute the complex-valued functions in real components, </p> cl ( x + i y ) = cl x − i sl x sl y cl y cl y + i sl x cl x sl y = cl x cl y ( 1 − sl 2 x sl 2 y ) cl 2 y + sl 2 x cl 2 x sl 2 y − i sl x sl y ( cl 2 x + cl 2 y ) cl 2 y + sl 2 x cl 2 x sl 2 y sl ( x + i y ) = sl x + i cl x sl y cl y cl y − i sl x cl x sl y = sl x cl y ( 1 − cl 2 x sl 2 y ) cl 2 y + sl 2 x cl 2 x sl 2 y + i cl x sl y ( sl 2 x + cl 2 y ) cl 2 y + sl 2 x cl 2 x sl 2 y {\displaystyle {\begin{aligned}\operatorname {cl} (x+iy)&={\frac {\operatorname {cl} x-i\operatorname {sl} x\,\operatorname {sl} y\,\operatorname {cl} y}{\operatorname {cl} y+i\operatorname {sl} x\,\operatorname {cl} x\,\operatorname {sl} y}}\\[4mu]&={\frac {\operatorname {cl} x\,\operatorname {cl} y\left(1-\operatorname {sl} ^{2}x\,\operatorname {sl} ^{2}y\right)}{\operatorname {cl} ^{2}y+\operatorname {sl} ^{2}x\,\operatorname {cl} ^{2}x\,\operatorname {sl} ^{2}y}}-i{\frac {\operatorname {sl} x\,\operatorname {sl} y\left(\operatorname {cl} ^{2}x+\operatorname {cl} ^{2}y\right)}{\operatorname {cl} ^{2}y+\operatorname {sl} ^{2}x\,\operatorname {cl} ^{2}x\,\operatorname {sl} ^{2}y}}\\[12mu]\operatorname {sl} (x+iy)&={\frac {\operatorname {sl} x+i\operatorname {cl} x\,\operatorname {sl} y\,\operatorname {cl} y}{\operatorname {cl} y-i\operatorname {sl} x\,\operatorname {cl} x\,\operatorname {sl} y}}\\[4mu]&={\frac {\operatorname {sl} x\,\operatorname {cl} y\left(1-\operatorname {cl} ^{2}x\,\operatorname {sl} ^{2}y\right)}{\operatorname {cl} ^{2}y+\operatorname {sl} ^{2}x\,\operatorname {cl} ^{2}x\,\operatorname {sl} ^{2}y}}+i{\frac {\operatorname {cl} x\,\operatorname {sl} y\left(\operatorname {sl} ^{2}x+\operatorname {cl} ^{2}y\right)}{\operatorname {cl} ^{2}y+\operatorname {sl} ^{2}x\,\operatorname {cl} ^{2}x\,\operatorname {sl} ^{2}y}}\end{aligned}}} <p>Gauss discovered that </p> sl ( u − v ) sl ( u + v ) = sl ( ( 1 + i ) u ) − sl ( ( 1 + i ) v ) sl ( ( 1 + i ) u ) + sl ( ( 1 + i ) v ) {\displaystyle {\frac {\operatorname {sl} (u-v)}{\operatorname {sl} (u+v)}}={\frac {\operatorname {sl} ((1+i)u)-\operatorname {sl} ((1+i)v)}{\operatorname {sl} ((1+i)u)+\operatorname {sl} ((1+i)v)}}} <p>where u , v ∈ C {\displaystyle u,v\in \mathbb {C} } such that both sides are well-defined. </p><p>Also </p> sl ( u + v ) sl ( u − v ) = sl 2 u − sl 2 v 1 + sl 2 u sl 2 v {\displaystyle \operatorname {sl} (u+v)\operatorname {sl} (u-v)={\frac {\operatorname {sl} ^{2}u-\operatorname {sl} ^{2}v}{1+\operatorname {sl} ^{2}u\operatorname {sl} ^{2}v}}} <p>where u , v ∈ C {\displaystyle u,v\in \mathbb {C} } such that both sides are well-defined; this resembles the trigonometric analog </p> sin ( u + v ) sin ( u − v ) = sin 2 u − sin 2 v . {\displaystyle \sin(u+v)\sin(u-v)=\sin ^{2}u-\sin ^{2}v.} <p>Bisection formulas: </p> cl 2 1 2 x = 1 + cl x 1 + sl 2 x 1 + sl 2 x + 1 {\displaystyle \operatorname {cl} ^{2}{\tfrac {1}{2}}x={\frac {1+\operatorname {cl} x{\sqrt {1+\operatorname {sl} ^{2}x}}}{{\sqrt {1+\operatorname {sl} ^{2}x}}+1}}} sl 2 1 2 x = 1 − cl x 1 + sl 2 x 1 + sl 2 x + 1 {\displaystyle \operatorname {sl} ^{2}{\tfrac {1}{2}}x={\frac {1-\operatorname {cl} x{\sqrt {1+\operatorname {sl} ^{2}x}}}{{\sqrt {1+\operatorname {sl} ^{2}x}}+1}}} <p>Duplication formulas:<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> cl 2 x = − 1 + 2 cl 2 x + cl 4 x 1 + 2 cl 2 x − cl 4 x {\displaystyle \operatorname {cl} 2x={\frac {-1+2\,\operatorname {cl} ^{2}x+\operatorname {cl} ^{4}x}{1+2\,\operatorname {cl} ^{2}x-\operatorname {cl} ^{4}x}}} sl 2 x = 2 sl x cl x 1 + sl 2 x 1 + sl 4 x {\displaystyle \operatorname {sl} 2x=2\,\operatorname {sl} x\,\operatorname {cl} x{\frac {1+\operatorname {sl} ^{2}x}{1+\operatorname {sl} ^{4}x}}} <p>Triplication formulas:<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> cl 3 x = − 3 cl x + 6 cl 5 x + cl 9 x 1 + 6 cl 4 x − 3 cl 8 x {\displaystyle \operatorname {cl} 3x={\frac {-3\,\operatorname {cl} x+6\,\operatorname {cl} ^{5}x+\operatorname {cl} ^{9}x}{1+6\,\operatorname {cl} ^{4}x-3\,\operatorname {cl} ^{8}x}}} sl 3 x = 3 sl x − 6 sl 5 x − 1 sl 9 x 1 + 6 sl 4 x − 3 sl 8 x {\displaystyle \operatorname {sl} 3x={\frac {\color {red}{3}\,\color {black}{\operatorname {sl} x-\,}\color {green}{6}\,\color {black}{\operatorname {sl} ^{5}x-\,}\color {blue}{1}\,\color {black}{\operatorname {sl} ^{9}x}}{\color {blue}{1}\,\color {black}{+\,}\,\color {green}{6}\,\color {black}{\operatorname {sl} ^{4}x-\,}\color {red}{3}\,\color {black}{\operatorname {sl} ^{8}x}}}} <p>Note the "reverse symmetry" of the coefficients of numerator and denominator of sl 3 x {\displaystyle \operatorname {sl} 3x} . This phenomenon can be observed in multiplication formulas for sl β x {\displaystyle \operatorname {sl} \beta x} where β = m + n i {\displaystyle \beta =m+ni} whenever m , n ∈ Z {\displaystyle m,n\in \mathbb {Z} } and m + n {\displaystyle m+n} is odd.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p> <h3>Lemnatomic polynomials</h3> <p>Let L {\displaystyle L} be the <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> </p> L = Z ( 1 + i ) ϖ + Z ( 1 − i ) ϖ . {\displaystyle L=\mathbb {Z} (1+i)\varpi +\mathbb {Z} (1-i)\varpi .} <p>Furthermore, let K = Q ( i ) {\displaystyle K=\mathbb {Q} (i)} , O = Z [ i ] {\displaystyle {\mathcal {O}}=\mathbb {Z} [i]} , z ∈ C {\displaystyle z\in \mathbb {C} } , β = m + i n {\displaystyle \beta =m+in} , γ = m ′ + i n ′ {\displaystyle \gamma =m'+in'} (where m , n , m ′ , n ′ ∈ Z {\displaystyle m,n,m',n'\in \mathbb {Z} } ), m + n {\displaystyle m+n} be odd, m ′ + n ′ {\displaystyle m'+n'} be odd, γ ≡ 1 mod 2 ( 1 + i ) {\displaystyle \gamma \equiv 1\,\operatorname {mod} \,2(1+i)} and sl β z = M β ( sl z ) {\displaystyle \operatorname {sl} \beta z=M_{\beta }(\operatorname {sl} z)} . Then </p> M β ( x ) = i ε x P β ( x 4 ) Q β ( x 4 ) {\displaystyle M_{\beta }(x)=i^{\varepsilon }x{\frac {P_{\beta }(x^{4})}{Q_{\beta }(x^{4})}}} <p>for some coprime polynomials P β ( x ) , Q β ( x ) ∈ O [ x ] {\displaystyle P_{\beta }(x),Q_{\beta }(x)\in {\mathcal {O}}[x]} and some ε ∈ { 0 , 1 , 2 , 3 } {\displaystyle \varepsilon \in \{0,1,2,3\}} <a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> where </p> x P β ( x 4 ) = ∏ γ | β Λ γ ( x ) {\displaystyle xP_{\beta }(x^{4})=\prod _{\gamma |\beta }\Lambda _{\gamma }(x)} <p>and </p> Λ β ( x ) = ∏ [ α ] ∈ ( O / β O ) × ( x − sl α δ β ) {\displaystyle \Lambda _{\beta }(x)=\prod _{[\alpha ]\in ({\mathcal {O}}/\beta {\mathcal {O}})^{\times }}(x-\operatorname {sl} \alpha \delta _{\beta })} <p>where δ β {\displaystyle \delta _{\beta }} is any β {\displaystyle \beta } -<a href="/facts/Torsion_(algebra)/uNRnwtLz">torsion</a> generator (i.e. δ β ∈ ( 1 / β ) L {\displaystyle \delta _{\beta }\in (1/\beta )L} and [ δ β ] ∈ ( 1 / β ) L / L {\displaystyle [\delta _{\beta }]\in (1/\beta )L/L} generates ( 1 / β ) L / L {\displaystyle (1/\beta )L/L} as an O {\displaystyle {\mathcal {O}}} -<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>). Examples of β {\displaystyle \beta } -torsion generators include 2 ϖ / β {\displaystyle 2\varpi /\beta } and ( 1 + i ) ϖ / β {\displaystyle (1+i)\varpi /\beta } . The polynomial Λ β ( x ) ∈ O [ x ] {\displaystyle \Lambda _{\beta }(x)\in {\mathcal {O}}[x]} is called the β {\displaystyle \beta } -th lemnatomic polynomial. It is monic and is irreducible over K {\displaystyle K} . The lemnatomic polynomials are the "lemniscate analogs" of the <a href="/facts/Cyclotomic_polynomials/TsWXDHIp">cyclotomic polynomials</a>,<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> </p> Φ k ( x ) = ∏ [ a ] ∈ ( Z / k Z ) × ( x − ζ k a ) . {\displaystyle \Phi _{k}(x)=\prod _{[a]\in (\mathbb {Z} /k\mathbb {Z} )^{\times }}(x-\zeta _{k}^{a}).} <p>The β {\displaystyle \beta } -th lemnatomic polynomial Λ β ( x ) {\displaystyle \Lambda _{\beta }(x)} is the <a href="/facts/Minimal_polynomial_(field_theory)/dJv9Q0Kh">minimal polynomial</a> of sl δ β {\displaystyle \operatorname {sl} \delta _{\beta }} in K [ x ] {\displaystyle K[x]} . For convenience, let ω β = sl ( 2 ϖ / β ) {\displaystyle \omega _{\beta }=\operatorname {sl} (2\varpi /\beta )} and ω ~ β = sl ( ( 1 + i ) ϖ / β ) {\displaystyle {\tilde {\omega }}_{\beta }=\operatorname {sl} ((1+i)\varpi /\beta )} . So for example, the minimal polynomial of ω 5 {\displaystyle \omega _{5}} (and also of ω ~ 5 {\displaystyle {\tilde {\omega }}_{5}} ) in K [ x ] {\displaystyle K[x]} is </p> Λ 5 ( x ) = x 16 + 52 x 12 − 26 x 8 − 12 x 4 + 1 , {\displaystyle \Lambda _{5}(x)=x^{16}+52x^{12}-26x^{8}-12x^{4}+1,} <p>and<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> ω 5 = − 13 + 6 5 + 2 85 − 38 5 4 {\displaystyle \omega _{5}={\sqrt[{4}]{-13+6{\sqrt {5}}+2{\sqrt {85-38{\sqrt {5}}}}}}} ω ~ 5 = − 13 − 6 5 + 2 85 + 38 5 4 {\displaystyle {\tilde {\omega }}_{5}={\sqrt[{4}]{-13-6{\sqrt {5}}+2{\sqrt {85+38{\sqrt {5}}}}}}} <a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> <p>(an equivalent expression is given in the table below). Another example is<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> </p> Λ − 1 + 2 i ( x ) = x 4 − 1 + 2 i {\displaystyle \Lambda _{-1+2i}(x)=x^{4}-1+2i} <p>which is the minimal polynomial of ω − 1 + 2 i {\displaystyle \omega _{-1+2i}} (and also of ω ~ − 1 + 2 i {\displaystyle {\tilde {\omega }}_{-1+2i}} ) in K [ x ] . {\displaystyle K[x].} </p><p>If p {\displaystyle p} is prime and β {\displaystyle \beta } is positive and odd,<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> then<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> </p> deg Λ β = β 2 ∏ p | β ( 1 − 1 p ) ( 1 − ( − 1 ) ( p − 1 ) / 2 p ) {\displaystyle \operatorname {deg} \Lambda _{\beta }=\beta ^{2}\prod _{p|\beta }\left(1-{\frac {1}{p}}\right)\left(1-{\frac {(-1)^{(p-1)/2}}{p}}\right)} <p>which can be compared to the cyclotomic analog </p> deg Φ k = k ∏ p | k ( 1 − 1 p ) . {\displaystyle \operatorname {deg} \Phi _{k}=k\prod _{p|k}\left(1-{\frac {1}{p}}\right).} <h3>Specific values</h3> <p>Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into <i>n</i> parts of equal length, using only basic arithmetic and square roots, if and only if <i>n</i> is of the form n = 2 k p 1 p 2 ⋯ p m {\displaystyle n=2^{k}p_{1}p_{2}\cdots p_{m}} where <i>k</i> is a non-negative <a href="/facts/Integer/PUwwrawx">integer</a> and each <i>p</i><i>i</i> (if any) is a distinct <a href="/facts/Fermat_prime/SJT8fN9m">Fermat prime</a>.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> </p> <table><tbody><tr><th> n {\displaystyle n} </th><th> cl n ϖ {\displaystyle \operatorname {cl} n\varpi } </th><th> sl n ϖ {\displaystyle \operatorname {sl} n\varpi } </th></tr><tr><td> 1 {\displaystyle 1} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 5 6 {\displaystyle {\tfrac {5}{6}}} </td><td> − 2 3 − 3 4 {\displaystyle -{\sqrt[{4}]{2{\sqrt {3}}-3}}} </td><td> 1 2 ( 3 + 1 − 12 4 ) {\displaystyle {\tfrac {1}{2}}{\bigl (}{\sqrt {3}}+1-{\sqrt[{4}]{12}}{\bigr )}} </td></tr><tr><td> 3 4 {\displaystyle {\tfrac {3}{4}}} </td><td> − 2 − 1 {\displaystyle -{\sqrt {{\sqrt {2}}-1}}} </td><td> 2 − 1 {\displaystyle {\sqrt {{\sqrt {2}}-1}}} </td></tr><tr><td> 2 3 {\displaystyle {\tfrac {2}{3}}} </td><td> − 1 2 ( 3 + 1 − 12 4 ) {\displaystyle -{\tfrac {1}{2}}{\bigl (}{\sqrt {3}}+1-{\sqrt[{4}]{12}}{\bigr )}} </td><td> 2 3 − 3 4 {\displaystyle {\sqrt[{4}]{2{\sqrt {3}}-3}}} </td></tr><tr><td> 1 2 {\displaystyle {\tfrac {1}{2}}} </td><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td></tr><tr><td> 1 3 {\displaystyle {\tfrac {1}{3}}} </td><td> 1 2 ( 3 + 1 − 12 4 ) {\displaystyle {\tfrac {1}{2}}{\bigl (}{\sqrt {3}}+1-{\sqrt[{4}]{12}}{\bigr )}} </td><td> 2 3 − 3 4 {\displaystyle {\sqrt[{4}]{2{\sqrt {3}}-3}}} </td></tr><tr><td> 1 4 {\displaystyle {\tfrac {1}{4}}} </td><td> 2 − 1 {\displaystyle {\sqrt {{\sqrt {2}}-1}}} </td><td> 2 − 1 {\displaystyle {\sqrt {{\sqrt {2}}-1}}} </td></tr><tr><td> 1 6 {\displaystyle {\tfrac {1}{6}}} </td><td> 2 3 − 3 4 {\displaystyle {\sqrt[{4}]{2{\sqrt {3}}-3}}} </td><td> 1 2 ( 3 + 1 − 12 4 ) {\displaystyle {\tfrac {1}{2}}{\bigl (}{\sqrt {3}}+1-{\sqrt[{4}]{12}}{\bigr )}} </td></tr></tbody></table> <h2 id="relation-to-geometric-shapes">Relation to geometric shapes</h2> <h3>Arc length of Bernoulli's lemniscate</h3> <p> L {\displaystyle {\mathcal {L}}} , the <a href="/facts/Lemniscate_of_Bernoulli/eIuuaQRh">lemniscate of Bernoulli</a> with unit distance from its center to its furthest point (i.e. with unit "half-width"), is essential in the theory of the lemniscate elliptic functions. It can be <a href="/facts/Characterization_(mathematics)/mmT1knFI">characterized</a> in at least three ways: </p><p>Angular characterization: Given two points A {\displaystyle A} and B {\displaystyle B} which are unit distance apart, let B ′ {\displaystyle B'} be the <a href="/facts/Point_reflection/UzswAm8N">reflection</a> of B {\displaystyle B} about A {\displaystyle A} . Then L {\displaystyle {\mathcal {L}}} is the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of the locus of the points P {\displaystyle P} such that | A P B − A P B ′ | {\displaystyle |APB-APB'|} is a <a href="/facts/Right_angle/psaslRq7">right angle</a>.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> </p><p>Focal characterization: L {\displaystyle {\mathcal {L}}} is the locus of points in the plane such that the product of their distances from the two focal points F 1 = ( − 1 2 , 0 ) {\displaystyle F_{1}={\bigl (}{-{\tfrac {1}{\sqrt {2}}}},0{\bigr )}} and F 2 = ( 1 2 , 0 ) {\displaystyle F_{2}={\bigl (}{\tfrac {1}{\sqrt {2}}},0{\bigr )}} is the constant 1 2 {\displaystyle {\tfrac {1}{2}}} . </p><p>Explicit coordinate characterization: L {\displaystyle {\mathcal {L}}} is a <a href="/facts/Quartic_plane_curve/5j35LcSS">quartic curve</a> satisfying the <a href="/facts/Polar_coordinate_system/HBko4YoN">polar</a> equation r 2 = cos 2 θ {\displaystyle r^{2}=\cos 2\theta } or the <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian</a> equation ( x 2 + y 2 ) 2 = x 2 − y 2 . {\displaystyle {\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}.} </p><p>The <a href="/facts/Perimeter/yQv1AajA">perimeter</a> of L {\displaystyle {\mathcal {L}}} is 2 ϖ {\displaystyle 2\varpi } .<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> </p><p>The points on L {\displaystyle {\mathcal {L}}} at distance r {\displaystyle r} from the origin are the intersections of the circle x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} and the <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> x 2 − y 2 = r 4 {\displaystyle x^{2}-y^{2}=r^{4}} . The intersection in the positive quadrant has Cartesian coordinates: </p> ( x ( r ) , y ( r ) ) = ( 1 2 r 2 ( 1 + r 2 ) , 1 2 r 2 ( 1 − r 2 ) ) . {\displaystyle {\big (}x(r),y(r){\big )}={\biggl (}\!{\sqrt {{\tfrac {1}{2}}r^{2}{\bigl (}1+r^{2}{\bigr )}}},\,{\sqrt {{\tfrac {1}{2}}r^{2}{\bigl (}1-r^{2}{\bigr )}}}\,{\biggr )}.} <p>Using this <a href="/facts/Parametric_equation/5Uvrlgqf">parametrization</a> with r ∈ [ 0 , 1 ] {\displaystyle r\in [0,1]} for a quarter of L {\displaystyle {\mathcal {L}}} , the <a href="/facts/Arc_length/QgTrmLxY">arc length</a> from the origin to a point ( x ( r ) , y ( r ) ) {\displaystyle {\big (}x(r),y(r){\big )}} is:<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p> ∫ 0 r x ′ ( t ) 2 + y ′ ( t ) 2 d t = ∫ 0 r ( 1 + 2 t 2 ) 2 2 ( 1 + t 2 ) + ( 1 − 2 t 2 ) 2 2 ( 1 − t 2 ) d t = ∫ 0 r d t 1 − t 4 = arcsl r . {\displaystyle {\begin{aligned}&\int _{0}^{r}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\mathop {\mathrm {d} t} \\&\quad {}=\int _{0}^{r}{\sqrt {{\frac {(1+2t^{2})^{2}}{2(1+t^{2})}}+{\frac {(1-2t^{2})^{2}}{2(1-t^{2})}}}}\mathop {\mathrm {d} t} \\[6mu]&\quad {}=\int _{0}^{r}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}\\[6mu]&\quad {}=\operatorname {arcsl} r.\end{aligned}}} <p>Likewise, the arc length from ( 1 , 0 ) {\displaystyle (1,0)} to ( x ( r ) , y ( r ) ) {\displaystyle {\big (}x(r),y(r){\big )}} is: </p> ∫ r 1 x ′ ( t ) 2 + y ′ ( t ) 2 d t = ∫ r 1 d t 1 − t 4 = arccl r = 1 2 ϖ − arcsl r . {\displaystyle {\begin{aligned}&\int _{r}^{1}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\mathop {\mathrm {d} t} \\&\quad {}=\int _{r}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}\\[6mu]&\quad {}=\operatorname {arccl} r={\tfrac {1}{2}}\varpi -\operatorname {arcsl} r.\end{aligned}}} <p>Or in the inverse direction, the lemniscate sine and cosine functions give the distance from the origin as functions of arc length from the origin and the point ( 1 , 0 ) {\displaystyle (1,0)} , respectively. </p><p>Analogously, the circular sine and cosine functions relate the chord length to the arc length for the unit diameter circle with polar equation r = cos θ {\displaystyle r=\cos \theta } or Cartesian equation x 2 + y 2 = x , {\displaystyle x^{2}+y^{2}=x,} using the same argument above but with the parametrization: </p> ( x ( r ) , y ( r ) ) = ( r 2 , r 2 ( 1 − r 2 ) ) . {\displaystyle {\big (}x(r),y(r){\big )}={\biggl (}r^{2},\,{\sqrt {r^{2}{\bigl (}1-r^{2}{\bigr )}}}\,{\biggr )}.} <p>Alternatively, just as the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} is parametrized in terms of the arc length s {\displaystyle s} from the point ( 1 , 0 ) {\displaystyle (1,0)} by </p> ( x ( s ) , y ( s ) ) = ( cos s , sin s ) , {\displaystyle (x(s),y(s))=(\cos s,\sin s),} <p> L {\displaystyle {\mathcal {L}}} is parametrized in terms of the arc length s {\displaystyle s} from the point ( 1 , 0 ) {\displaystyle (1,0)} by<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> </p> ( x ( s ) , y ( s ) ) = ( cl s 1 + sl 2 s , sl s cl s 1 + sl 2 s ) = ( cl ~ s , sl ~ s ) . {\displaystyle (x(s),y(s))=\left({\frac {\operatorname {cl} s}{\sqrt {1+\operatorname {sl} ^{2}s}}},{\frac {\operatorname {sl} s\operatorname {cl} s}{\sqrt {1+\operatorname {sl} ^{2}s}}}\right)=\left({\tilde {\operatorname {cl} }}\,s,{\tilde {\operatorname {sl} }}\,s\right).} <p>The notation cl ~ , sl ~ {\displaystyle {\tilde {\operatorname {cl} }},\,{\tilde {\operatorname {sl} }}} is used solely for the purposes of this article; in references, notation for general Jacobi elliptic functions is used instead. </p><p>The lemniscate integral and lemniscate functions satisfy an argument duplication identity discovered by Fagnano in 1718:<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> </p> ∫ 0 z d t 1 − t 4 = 2 ∫ 0 u d t 1 − t 4 , if z = 2 u 1 − u 4 1 + u 4 and 0 ≤ u ≤ 2 − 1 . {\displaystyle \int _{0}^{z}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}=2\int _{0}^{u}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}},\quad {\text{if }}z={\frac {2u{\sqrt {1-u^{4}}}}{1+u^{4}}}{\text{ and }}0\leq u\leq {\sqrt {{\sqrt {2}}-1}}.} <p>Later mathematicians generalized this result. Analogously to the <a href="/facts/Constructible_polygon/ATBRc55D">constructible polygons</a> in the circle, the lemniscate can be divided into <i>n</i> sections of equal arc length using only <a href="/facts/Straightedge_and_compass_construction/lbaHf2m7">straightedge and compass</a> if and only if <i>n</i> is of the form n = 2 k p 1 p 2 ⋯ p m {\displaystyle n=2^{k}p_{1}p_{2}\cdots p_{m}} where <i>k</i> is a non-negative <a href="/facts/Integer/PUwwrawx">integer</a> and each <i>p</i><i>i</i> (if any) is a distinct <a href="/facts/Fermat_prime/SJT8fN9m">Fermat prime</a>.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> The "if" part of the theorem was proved by <a href="/facts/Niels_Henrik_Abel/6kMfdRkT">Niels Abel</a> in 1827–1828, and the "only if" part was proved by <a href="/facts/Michael_Rosen_(mathematician)/63pUD7gC">Michael Rosen</a> in 1981.<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> Equivalently, the lemniscate can be divided into <i>n</i> sections of equal arc length using only straightedge and compass if and only if φ ( n ) {\displaystyle \varphi (n)} is a power of two (where φ {\displaystyle \varphi } is <a href="/facts/Euler%27s_totient_function/mIQwXYzj">Euler's totient function</a>). The lemniscate is <i>not</i> assumed to be already drawn, as that would go against the rules of straightedge and compass constructions; instead, it is assumed that we are given only two points by which the lemniscate is defined, such as its center and radial point (one of the two points on the lemniscate such that their distance from the center is maximal) or its two foci. </p><p>Let r j = sl 2 j ϖ n {\displaystyle r_{j}=\operatorname {sl} {\dfrac {2j\varpi }{n}}} . Then the <i>n</i>-division points for L {\displaystyle {\mathcal {L}}} are the points </p> ( r j 1 2 ( 1 + r j 2 ) , ( − 1 ) ⌊ 4 j / n ⌋ 1 2 r j 2 ( 1 − r j 2 ) ) , j ∈ { 1 , 2 , … , n } {\displaystyle \left(r_{j}{\sqrt {{\tfrac {1}{2}}{\bigl (}1+r_{j}^{2}{\bigr )}}},\ (-1)^{\left\lfloor 4j/n\right\rfloor }{\sqrt {{\tfrac {1}{2}}r_{j}^{2}{\bigl (}1-r_{j}^{2}{\bigr )}}}\right),\quad j\in \{1,2,\ldots ,n\}} <p>where ⌊ ⋅ ⌋ {\displaystyle \lfloor \cdot \rfloor } is the <a href="/facts/Floor_function/ssehyMMf">floor function</a>. See below for some specific values of sl 2 ϖ n {\displaystyle \operatorname {sl} {\dfrac {2\varpi }{n}}} . </p> <h3>Arc length of rectangular elastica</h3> <p>The inverse lemniscate sine also describes the arc length <i>s</i> relative to the <i>x</i> coordinate of the rectangular <a href="/facts/Elastica_theory/B0W70h8a">elastica</a>.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> This curve has <i>y</i> coordinate and arc length: </p> y = ∫ x 1 t 2 d t 1 − t 4 , s = arcsl x = ∫ 0 x d t 1 − t 4 {\displaystyle y=\int _{x}^{1}{\frac {t^{2}\mathop {\mathrm {d} t} }{\sqrt {1-t^{4}}}},\quad s=\operatorname {arcsl} x=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}} <p>The rectangular elastica solves a problem posed by <a href="/facts/Jacob_Bernoulli/rUnwozsy">Jacob Bernoulli</a>, in 1691, to describe the shape of an idealized flexible rod fixed in a vertical orientation at the bottom end, and pulled down by a weight from the far end until it has been bent horizontal. Bernoulli's proposed solution established <a href="/facts/Euler%E2%80%93Bernoulli_beam_theory/3DyQV9do">Euler–Bernoulli beam theory</a>, further developed by Euler in the 18th century. </p> <h3>Elliptic characterization</h3> <p>Let C {\displaystyle C} be a point on the ellipse x 2 + 2 y 2 = 1 {\displaystyle x^{2}+2y^{2}=1} in the first quadrant and let D {\displaystyle D} be the projection of C {\displaystyle C} on the unit circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} . The distance r {\displaystyle r} between the origin A {\displaystyle A} and the point C {\displaystyle C} is a function of φ {\displaystyle \varphi } (the angle B A C {\displaystyle BAC} where B = ( 1 , 0 ) {\displaystyle B=(1,0)} ; equivalently the length of the circular arc B D {\displaystyle BD} ). The parameter u {\displaystyle u} is given by </p> u = ∫ 0 φ r ( θ ) d θ = ∫ 0 φ d θ 1 + sin 2 θ . {\displaystyle u=\int _{0}^{\varphi }r(\theta )\,\mathrm {d} \theta =\int _{0}^{\varphi }{\frac {\mathrm {d} \theta }{\sqrt {1+\sin ^{2}\theta }}}.} <p>If E {\displaystyle E} is the projection of D {\displaystyle D} on the x-axis and if F {\displaystyle F} is the projection of C {\displaystyle C} on the x-axis, then the lemniscate elliptic functions are given by </p> cl u = A F ¯ , sl u = D E ¯ , {\displaystyle \operatorname {cl} u={\overline {AF}},\quad \operatorname {sl} u={\overline {DE}},} cl ~ u = A F ¯ A C ¯ , sl ~ u = A F ¯ F C ¯ . {\displaystyle {\tilde {\operatorname {cl} }}\,u={\overline {AF}}{\overline {AC}},\quad {\tilde {\operatorname {sl} }}\,u={\overline {AF}}{\overline {FC}}.} <h2 id="series-identities">Series Identities</h2> <h3>Power series</h3> <p>The <a href="/facts/Power_series/vYh6sTps">power series</a> expansion of the lemniscate sine at the origin is<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> </p> sl z = ∑ n = 0 ∞ a n z n = z − 12 z 5 5 ! + 3024 z 9 9 ! − 4390848 z 13 13 ! + ⋯ , | z | < ϖ 2 {\displaystyle \operatorname {sl} z=\sum _{n=0}^{\infty }a_{n}z^{n}=z-12{\frac {z^{5}}{5!}}+3024{\frac {z^{9}}{9!}}-4390848{\frac {z^{13}}{13!}}+\cdots ,\quad |z|<{\tfrac {\varpi }{\sqrt {2}}}} <p>where the coefficients a n {\displaystyle a_{n}} are determined as follows: </p> n ≢ 1 ( mod 4 ) ⟹ a n = 0 , {\displaystyle n\not \equiv 1{\pmod {4}}\implies a_{n}=0,} a 1 = 1 , ∀ n ∈ N 0 : a n + 2 = − 2 ( n + 1 ) ( n + 2 ) ∑ i + j + k = n a i a j a k {\displaystyle a_{1}=1,\,\forall n\in \mathbb {N} _{0}:\,a_{n+2}=-{\frac {2}{(n+1)(n+2)}}\sum _{i+j+k=n}a_{i}a_{j}a_{k}} <p>where i + j + k = n {\displaystyle i+j+k=n} stands for all three-term <a href="/facts/Composition_(combinatorics)/sfBgx7Cd">compositions</a> of n {\displaystyle n} . For example, to evaluate a 13 {\displaystyle a_{13}} , it can be seen that there are only six compositions of 13 − 2 = 11 {\displaystyle 13-2=11} that give a nonzero contribution to the sum: 11 = 9 + 1 + 1 = 1 + 9 + 1 = 1 + 1 + 9 {\displaystyle 11=9+1+1=1+9+1=1+1+9} and 11 = 5 + 5 + 1 = 5 + 1 + 5 = 1 + 5 + 5 {\displaystyle 11=5+5+1=5+1+5=1+5+5} , so </p> a 13 = − 2 12 ⋅ 13 ( a 9 a 1 a 1 + a 1 a 9 a 1 + a 1 a 1 a 9 + a 5 a 5 a 1 + a 5 a 1 a 5 + a 1 a 5 a 5 ) = − 11 15600 . {\displaystyle a_{13}=-{\tfrac {2}{12\cdot 13}}(a_{9}a_{1}a_{1}+a_{1}a_{9}a_{1}+a_{1}a_{1}a_{9}+a_{5}a_{5}a_{1}+a_{5}a_{1}a_{5}+a_{1}a_{5}a_{5})=-{\tfrac {11}{15600}}.} <p>The expansion can be equivalently written as<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a> </p> sl z = ∑ n = 0 ∞ p 2 n z 4 n + 1 ( 4 n + 1 ) ! , | z | < ϖ 2 {\displaystyle \operatorname {sl} z=\sum _{n=0}^{\infty }p_{2n}{\frac {z^{4n+1}}{(4n+1)!}},\quad \left|z\right|<{\frac {\varpi }{\sqrt {2}}}} <p>where </p> p n + 2 = − 12 ∑ j = 0 n ( 2 n + 2 2 j + 2 ) p n − j ∑ k = 0 j ( 2 j + 1 2 k + 1 ) p k p j − k , p 0 = 1 , p 1 = 0. {\displaystyle p_{n+2}=-12\sum _{j=0}^{n}{\binom {2n+2}{2j+2}}p_{n-j}\sum _{k=0}^{j}{\binom {2j+1}{2k+1}}p_{k}p_{j-k},\quad p_{0}=1,\,p_{1}=0.} <p>The power series expansion of sl ~ {\displaystyle {\tilde {\operatorname {sl} }}} at the origin is </p> sl ~ z = ∑ n = 0 ∞ α n z n = z − 9 z 3 3 ! + 153 z 5 5 ! − 4977 z 7 7 ! + ⋯ , | z | < ϖ 2 {\displaystyle {\tilde {\operatorname {sl} }}\,z=\sum _{n=0}^{\infty }\alpha _{n}z^{n}=z-9{\frac {z^{3}}{3!}}+153{\frac {z^{5}}{5!}}-4977{\frac {z^{7}}{7!}}+\cdots ,\quad \left|z\right|<{\frac {\varpi }{2}}} <p>where α n = 0 {\displaystyle \alpha _{n}=0} if n {\displaystyle n} is even and<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a> </p> α n = 2 π ϖ ( − 1 ) ( n − 1 ) / 2 n ! ∑ k = 1 ∞ ( 2 k π / ϖ ) n + 1 cosh k π , | α n | ∼ 2 n + 5 / 2 n + 1 ϖ n + 2 {\displaystyle \alpha _{n}={\sqrt {2}}{\frac {\pi }{\varpi }}{\frac {(-1)^{(n-1)/2}}{n!}}\sum _{k=1}^{\infty }{\frac {(2k\pi /\varpi )^{n+1}}{\cosh k\pi }},\quad \left|\alpha _{n}\right|\sim 2^{n+5/2}{\frac {n+1}{\varpi ^{n+2}}}} <p>if n {\displaystyle n} is odd. </p><p>The expansion can be equivalently written as<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a> </p> sl ~ z = ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 ( ∑ l = 0 n 2 l ( 2 n + 2 2 l + 1 ) s l t n − l ) z 2 n + 1 ( 2 n + 1 ) ! , | z | < ϖ 2 {\displaystyle {\tilde {\operatorname {sl} }}\,z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{n+1}}}\left(\sum _{l=0}^{n}2^{l}{\binom {2n+2}{2l+1}}s_{l}t_{n-l}\right){\frac {z^{2n+1}}{(2n+1)!}},\quad \left|z\right|<{\frac {\varpi }{2}}} <p>where </p> s n + 2 = 3 s n + 1 + 24 ∑ j = 0 n ( 2 n + 2 2 j + 2 ) s n − j ∑ k = 0 j ( 2 j + 1 2 k + 1 ) s k s j − k , s 0 = 1 , s 1 = 3 , {\displaystyle s_{n+2}=3s_{n+1}+24\sum _{j=0}^{n}{\binom {2n+2}{2j+2}}s_{n-j}\sum _{k=0}^{j}{\binom {2j+1}{2k+1}}s_{k}s_{j-k},\quad s_{0}=1,\,s_{1}=3,} t n + 2 = 3 t n + 1 + 3 ∑ j = 0 n ( 2 n + 2 2 j + 2 ) t n − j ∑ k = 0 j ( 2 j + 1 2 k + 1 ) t k t j − k , t 0 = 1 , t 1 = 3. {\displaystyle t_{n+2}=3t_{n+1}+3\sum _{j=0}^{n}{\binom {2n+2}{2j+2}}t_{n-j}\sum _{k=0}^{j}{\binom {2j+1}{2k+1}}t_{k}t_{j-k},\quad t_{0}=1,\,t_{1}=3.} <p>For the lemniscate cosine,<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> </p> cl z = 1 − ∑ n = 0 ∞ ( − 1 ) n ( ∑ l = 0 n 2 l ( 2 n + 2 2 l + 1 ) q l r n − l ) z 2 n + 2 ( 2 n + 2 ) ! = 1 − 2 z 2 2 ! + 12 z 4 4 ! − 216 z 6 6 ! + ⋯ , | z | < ϖ 2 , {\displaystyle \operatorname {cl} {z}=1-\sum _{n=0}^{\infty }(-1)^{n}\left(\sum _{l=0}^{n}2^{l}{\binom {2n+2}{2l+1}}q_{l}r_{n-l}\right){\frac {z^{2n+2}}{(2n+2)!}}=1-2{\frac {z^{2}}{2!}}+12{\frac {z^{4}}{4!}}-216{\frac {z^{6}}{6!}}+\cdots ,\quad \left|z\right|<{\frac {\varpi }{2}},} cl ~ z = ∑ n = 0 ∞ ( − 1 ) n 2 n q n z 2 n ( 2 n ) ! = 1 − 3 z 2 2 ! + 33 z 4 4 ! − 819 z 6 6 ! + ⋯ , | z | < ϖ 2 {\displaystyle {\tilde {\operatorname {cl} }}\,z=\sum _{n=0}^{\infty }(-1)^{n}2^{n}q_{n}{\frac {z^{2n}}{(2n)!}}=1-3{\frac {z^{2}}{2!}}+33{\frac {z^{4}}{4!}}-819{\frac {z^{6}}{6!}}+\cdots ,\quad \left|z\right|<{\frac {\varpi }{2}}} <p>where </p> r n + 2 = 3 ∑ j = 0 n ( 2 n + 2 2 j + 2 ) r n − j ∑ k = 0 j ( 2 j + 1 2 k + 1 ) r k r j − k , r 0 = 1 , r 1 = 0 , {\displaystyle r_{n+2}=3\sum _{j=0}^{n}{\binom {2n+2}{2j+2}}r_{n-j}\sum _{k=0}^{j}{\binom {2j+1}{2k+1}}r_{k}r_{j-k},\quad r_{0}=1,\,r_{1}=0,} q n + 2 = 3 2 q n + 1 + 6 ∑ j = 0 n ( 2 n + 2 2 j + 2 ) q n − j ∑ k = 0 j ( 2 j + 1 2 k + 1 ) q k q j − k , q 0 = 1 , q 1 = 3 2 . {\displaystyle q_{n+2}={\tfrac {3}{2}}q_{n+1}+6\sum _{j=0}^{n}{\binom {2n+2}{2j+2}}q_{n-j}\sum _{k=0}^{j}{\binom {2j+1}{2k+1}}q_{k}q_{j-k},\quad q_{0}=1,\,q_{1}={\tfrac {3}{2}}.} <h3>Ramanujan's cos/cosh identity</h3> <p>Ramanujan's famous cos/cosh identity states that if </p> R ( s ) = π ϖ 2 ∑ n ∈ Z cos ( 2 n π s / ϖ ) cosh n π , {\displaystyle R(s)={\frac {\pi }{\varpi {\sqrt {2}}}}\sum _{n\in \mathbb {Z} }{\frac {\cos(2n\pi s/\varpi )}{\cosh n\pi }},} <p>then<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> </p> R ( s ) − 2 + R ( i s ) − 2 = 2 , | Re s | < ϖ 2 , | Im s | < ϖ 2 . {\displaystyle R(s)^{-2}+R(is)^{-2}=2,\quad \left|\operatorname {Re} s\right|<{\frac {\varpi }{2}},\left|\operatorname {Im} s\right|<{\frac {\varpi }{2}}.} <p>There is a close relation between the lemniscate functions and R ( s ) {\displaystyle R(s)} . Indeed,<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a><a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> </p> sl ~ s = − d d s R ( s ) | Im s | < ϖ 2 {\displaystyle {\tilde {\operatorname {sl} }}\,s=-{\frac {\mathrm {d} }{\mathrm {d} s}}R(s)\quad \left|\operatorname {Im} s\right|<{\frac {\varpi }{2}}} cl ~ s = d d s 1 − R ( s ) 2 , | Re s − ϖ 2 | < ϖ 2 , | Im s | < ϖ 2 {\displaystyle {\tilde {\operatorname {cl} }}\,s={\frac {\mathrm {d} }{\mathrm {d} s}}{\sqrt {1-R(s)^{2}}},\quad \left|\operatorname {Re} s-{\frac {\varpi }{2}}\right|<{\frac {\varpi }{2}},\,\left|\operatorname {Im} s\right|<{\frac {\varpi }{2}}} <p>and </p> R ( s ) = 1 1 + sl 2 s , | Im s | < ϖ 2 . {\displaystyle R(s)={\frac {1}{\sqrt {1+\operatorname {sl} ^{2}s}}},\quad \left|\operatorname {Im} s\right|<{\frac {\varpi }{2}}.} <h3>Continued fractions</h3> <p>For z ∈ C ∖ { 0 } {\displaystyle z\in \mathbb {C} \setminus \{0\}} :<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a> </p> ∫ 0 ∞ e − t z 2 cl t d t = 1 / 2 z + a 1 z + a 2 z + a 3 z + ⋱ , a n = n 2 4 ( ( − 1 ) n + 1 + 3 ) {\displaystyle \int _{0}^{\infty }e^{-tz{\sqrt {2}}}\operatorname {cl} t\,\mathrm {d} t={\cfrac {1/{\sqrt {2}}}{z+{\cfrac {a_{1}}{z+{\cfrac {a_{2}}{z+{\cfrac {a_{3}}{z+\ddots }}}}}}}},\quad a_{n}={\frac {n^{2}}{4}}((-1)^{n+1}+3)} ∫ 0 ∞ e − t z 2 sl t cl t d t = 1 / 2 z 2 + b 1 − a 1 z 2 + b 2 − a 2 z 2 + b 3 − ⋱ , a n = n 2 ( 4 n 2 − 1 ) , b n = 3 ( 2 n − 1 ) 2 {\displaystyle \int _{0}^{\infty }e^{-tz{\sqrt {2}}}\operatorname {sl} t\operatorname {cl} t\,\mathrm {d} t={\cfrac {1/2}{z^{2}+b_{1}-{\cfrac {a_{1}}{z^{2}+b_{2}-{\cfrac {a_{2}}{z^{2}+b_{3}-\ddots }}}}}},\quad a_{n}=n^{2}(4n^{2}-1),\,b_{n}=3(2n-1)^{2}} <h3>Methods of computation</h3> <p>Several methods of computing sl x {\displaystyle \operatorname {sl} x} involve first making the change of variables π x = ϖ x ~ {\displaystyle \pi x=\varpi {\tilde {x}}} and then computing sl ( ϖ x ~ / π ) . {\displaystyle \operatorname {sl} (\varpi {\tilde {x}}/\pi ).} </p><p>A <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic</a> series method:<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a><a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> </p> sl ( ϖ π x ) = π ϖ ∑ n ∈ Z ( − 1 ) n cosh ( x − ( n + 1 / 2 ) π ) , x ∈ C {\displaystyle \operatorname {sl} \left({\frac {\varpi }{\pi }}x\right)={\frac {\pi }{\varpi }}\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{\cosh(x-(n+1/2)\pi )}},\quad x\in \mathbb {C} } 1 sl ( ϖ x / π ) = π ϖ ∑ n ∈ Z ( − 1 ) n sinh ( x − n π ) = π ϖ ∑ n ∈ Z ( − 1 ) n sin ( x − n π i ) , x ∈ C {\displaystyle {\frac {1}{\operatorname {sl} (\varpi x/\pi )}}={\frac {\pi }{\varpi }}\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{{\sinh }{\left(x-n\pi \right)}}}={\frac {\pi }{\varpi }}\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{\sin(x-n\pi i)}},\quad x\in \mathbb {C} } <p><a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> method:<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a> </p> sl ( ϖ π x ) = 2 π ϖ ∑ n = 0 ∞ ( − 1 ) n sin ( ( 2 n + 1 ) x ) cosh ( ( n + 1 / 2 ) π ) , | Im x | < π 2 {\displaystyle \operatorname {sl} {\Bigl (}{\frac {\varpi }{\pi }}x{\Bigr )}={\frac {2\pi }{\varpi }}\sum _{n=0}^{\infty }{\frac {(-1)^{n}\sin((2n+1)x)}{\cosh((n+1/2)\pi )}},\quad \left|\operatorname {Im} x\right|<{\frac {\pi }{2}}} cl ( ϖ π x ) = 2 π ϖ ∑ n = 0 ∞ cos ( ( 2 n + 1 ) x ) cosh ( ( n + 1 / 2 ) π ) , | Im x | < π 2 {\displaystyle \operatorname {cl} \left({\frac {\varpi }{\pi }}x\right)={\frac {2\pi }{\varpi }}\sum _{n=0}^{\infty }{\frac {\cos((2n+1)x)}{\cosh((n+1/2)\pi )}},\quad \left|\operatorname {Im} x\right|<{\frac {\pi }{2}}} 1 sl ( ϖ x / π ) = π ϖ ( 1 sin x − 4 ∑ n = 0 ∞ sin ( ( 2 n + 1 ) x ) e ( 2 n + 1 ) π + 1 ) , | Im x | < π {\displaystyle {\frac {1}{\operatorname {sl} (\varpi x/\pi )}}={\frac {\pi }{\varpi }}\left({\frac {1}{\sin x}}-4\sum _{n=0}^{\infty }{\frac {\sin((2n+1)x)}{e^{(2n+1)\pi }+1}}\right),\quad \left|\operatorname {Im} x\right|<\pi } <p>The lemniscate functions can be computed more rapidly by </p> sl ( ϖ π x ) = θ 1 ( x , e − π ) θ 3 ( x , e − π ) , x ∈ C cl ( ϖ π x ) = θ 2 ( x , e − π ) θ 4 ( x , e − π ) , x ∈ C {\displaystyle {\begin{aligned}\operatorname {sl} {\Bigl (}{\frac {\varpi }{\pi }}x{\Bigr )}&={\frac {{\theta _{1}}{\left(x,e^{-\pi }\right)}}{{\theta _{3}}{\left(x,e^{-\pi }\right)}}},\quad x\in \mathbb {C} \\\operatorname {cl} {\Bigl (}{\frac {\varpi }{\pi }}x{\Bigr )}&={\frac {{\theta _{2}}{\left(x,e^{-\pi }\right)}}{{\theta _{4}}{\left(x,e^{-\pi }\right)}}},\quad x\in \mathbb {C} \end{aligned}}} <p>where </p> θ 1 ( x , e − π ) = ∑ n ∈ Z ( − 1 ) n + 1 e − π ( n + 1 / 2 + x / π ) 2 = ∑ n ∈ Z ( − 1 ) n e − π ( n + 1 / 2 ) 2 sin ( ( 2 n + 1 ) x ) , θ 2 ( x , e − π ) = ∑ n ∈ Z ( − 1 ) n e − π ( n + x / π ) 2 = ∑ n ∈ Z e − π ( n + 1 / 2 ) 2 cos ( ( 2 n + 1 ) x ) , θ 3 ( x , e − π ) = ∑ n ∈ Z e − π ( n + x / π ) 2 = ∑ n ∈ Z e − π n 2 cos 2 n x , θ 4 ( x , e − π ) = ∑ n ∈ Z e − π ( n + 1 / 2 + x / π ) 2 = ∑ n ∈ Z ( − 1 ) n e − π n 2 cos 2 n x {\displaystyle {\begin{aligned}\theta _{1}(x,e^{-\pi })&=\sum _{n\in \mathbb {Z} }(-1)^{n+1}e^{-\pi (n+1/2+x/\pi )^{2}}=\sum _{n\in \mathbb {Z} }(-1)^{n}e^{-\pi (n+1/2)^{2}}\sin((2n+1)x),\\\theta _{2}(x,e^{-\pi })&=\sum _{n\in \mathbb {Z} }(-1)^{n}e^{-\pi (n+x/\pi )^{2}}=\sum _{n\in \mathbb {Z} }e^{-\pi (n+1/2)^{2}}\cos((2n+1)x),\\\theta _{3}(x,e^{-\pi })&=\sum _{n\in \mathbb {Z} }e^{-\pi (n+x/\pi )^{2}}=\sum _{n\in \mathbb {Z} }e^{-\pi n^{2}}\cos 2nx,\\\theta _{4}(x,e^{-\pi })&=\sum _{n\in \mathbb {Z} }e^{-\pi (n+1/2+x/\pi )^{2}}=\sum _{n\in \mathbb {Z} }(-1)^{n}e^{-\pi n^{2}}\cos 2nx\end{aligned}}} <p>are the <a href="/facts/Theta_function/g09xvoQQ">Jacobi theta functions</a>.<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a> </p><p>Fourier series for the logarithm of the lemniscate sine: </p> ln sl ( ϖ π x ) = ln 2 − π 4 + ln sin x + 2 ∑ n = 1 ∞ ( − 1 ) n cos 2 n x n ( e n π + ( − 1 ) n ) , | Im x | < π 2 {\displaystyle \ln \operatorname {sl} \left({\frac {\varpi }{\pi }}x\right)=\ln 2-{\frac {\pi }{4}}+\ln \sin x+2\sum _{n=1}^{\infty }{\frac {(-1)^{n}\cos 2nx}{n(e^{n\pi }+(-1)^{n})}},\quad \left|\operatorname {Im} x\right|<{\frac {\pi }{2}}} <p>The following series identities were discovered by <a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Ramanujan</a>:<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a> </p> ϖ 2 π 2 sl 2 ( ϖ x / π ) = 1 sin 2 x − 1 π − 8 ∑ n = 1 ∞ n cos 2 n x e 2 n π − 1 , | Im x | < π {\displaystyle {\frac {\varpi ^{2}}{\pi ^{2}\operatorname {sl} ^{2}(\varpi x/\pi )}}={\frac {1}{\sin ^{2}x}}-{\frac {1}{\pi }}-8\sum _{n=1}^{\infty }{\frac {n\cos 2nx}{e^{2n\pi }-1}},\quad \left|\operatorname {Im} x\right|<\pi } arctan sl ( ϖ π x ) = 2 ∑ n = 0 ∞ sin ( ( 2 n + 1 ) x ) ( 2 n + 1 ) cosh ( ( n + 1 / 2 ) π ) , | Im x | < π 2 {\displaystyle \arctan \operatorname {sl} {\Bigl (}{\frac {\varpi }{\pi }}x{\Bigr )}=2\sum _{n=0}^{\infty }{\frac {\sin((2n+1)x)}{(2n+1)\cosh((n+1/2)\pi )}},\quad \left|\operatorname {Im} x\right|<{\frac {\pi }{2}}} <p>The functions sl ~ {\displaystyle {\tilde {\operatorname {sl} }}} and cl ~ {\displaystyle {\tilde {\operatorname {cl} }}} analogous to sin {\displaystyle \sin } and cos {\displaystyle \cos } on the unit circle have the following Fourier and hyperbolic series expansions:<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a><a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a> </p> sl ~ s = 2 2 π 2 ϖ 2 ∑ n = 1 ∞ n sin ( 2 n π s / ϖ ) cosh n π , | Im s | < ϖ 2 {\displaystyle {\tilde {\operatorname {sl} }}\,s=2{\sqrt {2}}{\frac {\pi ^{2}}{\varpi ^{2}}}\sum _{n=1}^{\infty }{\frac {n\sin(2n\pi s/\varpi )}{\cosh n\pi }},\quad \left|\operatorname {Im} s\right|<{\frac {\varpi }{2}}} cl ~ s = 2 π 2 ϖ 2 ∑ n = 0 ∞ ( 2 n + 1 ) cos ( ( 2 n + 1 ) π s / ϖ ) sinh ( ( n + 1 / 2 ) π ) , | Im s | < ϖ 2 {\displaystyle {\tilde {\operatorname {cl} }}\,s={\sqrt {2}}{\frac {\pi ^{2}}{\varpi ^{2}}}\sum _{n=0}^{\infty }{\frac {(2n+1)\cos((2n+1)\pi s/\varpi )}{\sinh((n+1/2)\pi )}},\quad \left|\operatorname {Im} s\right|<{\frac {\varpi }{2}}} sl ~ s = π 2 ϖ 2 2 ∑ n ∈ Z sinh ( π ( n + s / ϖ ) ) cosh 2 ( π ( n + s / ϖ ) ) , s ∈ C {\displaystyle {\tilde {\operatorname {sl} }}\,s={\frac {\pi ^{2}}{\varpi ^{2}{\sqrt {2}}}}\sum _{n\in \mathbb {Z} }{\frac {\sinh(\pi (n+s/\varpi ))}{\cosh ^{2}(\pi (n+s/\varpi ))}},\quad s\in \mathbb {C} } cl ~ s = π 2 ϖ 2 2 ∑ n ∈ Z ( − 1 ) n cosh 2 ( π ( n + s / ϖ ) ) , s ∈ C {\displaystyle {\tilde {\operatorname {cl} }}\,s={\frac {\pi ^{2}}{\varpi ^{2}{\sqrt {2}}}}\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{\cosh ^{2}(\pi (n+s/\varpi ))}},\quad s\in \mathbb {C} } <p>The following identities come from product representations of the theta functions:<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a> </p> s l ( ϖ π x ) = 2 e − π / 4 sin x ∏ n = 1 ∞ 1 − 2 e − 2 n π cos 2 x + e − 4 n π 1 + 2 e − ( 2 n − 1 ) π cos 2 x + e − ( 4 n − 2 ) π , x ∈ C {\displaystyle \mathrm {sl} {\Bigl (}{\frac {\varpi }{\pi }}x{\Bigr )}=2e^{-\pi /4}\sin x\prod _{n=1}^{\infty }{\frac {1-2e^{-2n\pi }\cos 2x+e^{-4n\pi }}{1+2e^{-(2n-1)\pi }\cos 2x+e^{-(4n-2)\pi }}},\quad x\in \mathbb {C} } c l ( ϖ π x ) = 2 e − π / 4 cos x ∏ n = 1 ∞ 1 + 2 e − 2 n π cos 2 x + e − 4 n π 1 − 2 e − ( 2 n − 1 ) π cos 2 x + e − ( 4 n − 2 ) π , x ∈ C {\displaystyle \mathrm {cl} {\Bigl (}{\frac {\varpi }{\pi }}x{\Bigr )}=2e^{-\pi /4}\cos x\prod _{n=1}^{\infty }{\frac {1+2e^{-2n\pi }\cos 2x+e^{-4n\pi }}{1-2e^{-(2n-1)\pi }\cos 2x+e^{-(4n-2)\pi }}},\quad x\in \mathbb {C} } <p>A similar formula involving the sn {\displaystyle \operatorname {sn} } function can be given.<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a> </p> <h3>The lemniscate functions as a ratio of entire functions</h3> <p>Since the lemniscate sine is a meromorphic function in the whole complex plane, it can be written as a ratio of <a href="/facts/Entire_function/XNluKzu7">entire functions</a>. Gauss showed that sl has the following product expansion, reflecting the distribution of its zeros and poles:<a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a> </p> sl z = M ( z ) N ( z ) {\displaystyle \operatorname {sl} z={\frac {M(z)}{N(z)}}} <p>where </p> M ( z ) = z ∏ α ( 1 − z 4 α 4 ) , N ( z ) = ∏ β ( 1 − z 4 β 4 ) . {\displaystyle M(z)=z\prod _{\alpha }\left(1-{\frac {z^{4}}{\alpha ^{4}}}\right),\quad N(z)=\prod _{\beta }\left(1-{\frac {z^{4}}{\beta ^{4}}}\right).} <p>Here, α {\displaystyle \alpha } and β {\displaystyle \beta } denote, respectively, the zeros and poles of sl which are in the quadrant Re z > 0 , Im z ≥ 0 {\displaystyle \operatorname {Re} z>0,\operatorname {Im} z\geq 0} . A proof can be found in.<a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a><a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a> Importantly, the infinite products converge to the same value for all possible orders in which their terms can be multiplied, as a consequence of <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a>.<a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a> </p> <table><tbody><tr><th>Proof of the infinite product for the lemniscate sine</th></tr><tr><td><p>Proof by logarithmic differentiation</p><p>It can be easily seen (using uniform and <a href="/facts/Absolute_convergence/BXz9kBJ5">absolute</a> convergence arguments to justify <a href="/facts/Interchange_of_limiting_operations/qqW0RnEn">interchanging of limiting operations</a>) that</p> M ′ ( z ) M ( z ) = − ∑ n = 0 ∞ 2 4 n H 4 n z 4 n − 1 ( 4 n ) ! , | z | < ϖ {\displaystyle {\frac {M'(z)}{M(z)}}=-\sum _{n=0}^{\infty }2^{4n}\mathrm {H} _{4n}{\frac {z^{4n-1}}{(4n)!}},\quad \left|z\right|<\varpi } <p>(where H n {\displaystyle \mathrm {H} _{n}} are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers) and</p> N ′ ( z ) N ( z ) = ( 1 + i ) M ′ ( ( 1 + i ) z ) M ( ( 1 + i ) z ) − M ′ ( z ) M ( z ) . {\displaystyle {\frac {N'(z)}{N(z)}}=(1+i){\frac {M'((1+i)z)}{M((1+i)z)}}-{\frac {M'(z)}{M(z)}}.} <p>Therefore</p> N ′ ( z ) N ( z ) = ∑ n = 0 ∞ 2 4 n ( 1 − ( − 1 ) n 2 2 n ) H 4 n z 4 n − 1 ( 4 n ) ! , | z | < ϖ 2 . {\displaystyle {\frac {N'(z)}{N(z)}}=\sum _{n=0}^{\infty }2^{4n}(1-(-1)^{n}2^{2n})\mathrm {H} _{4n}{\frac {z^{4n-1}}{(4n)!}},\quad \left|z\right|<{\frac {\varpi }{\sqrt {2}}}.} <p>It is known that</p> 1 sl 2 z = ∑ n = 0 ∞ 2 4 n ( 4 n − 1 ) H 4 n z 4 n − 2 ( 4 n ) ! , | z | < ϖ . {\displaystyle {\frac {1}{\operatorname {sl} ^{2}z}}=\sum _{n=0}^{\infty }2^{4n}(4n-1)\mathrm {H} _{4n}{\frac {z^{4n-2}}{(4n)!}},\quad \left|z\right|<\varpi .} <p>Then from</p> d d z sl ′ z sl z = − 1 sl 2 z − sl 2 z {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}{\frac {\operatorname {sl} 'z}{\operatorname {sl} z}}=-{\frac {1}{\operatorname {sl} ^{2}z}}-\operatorname {sl} ^{2}z} <p>and</p> sl 2 z = 1 sl 2 z − ( 1 + i ) 2 sl 2 ( ( 1 + i ) z ) {\displaystyle \operatorname {sl} ^{2}z={\frac {1}{\operatorname {sl} ^{2}z}}-{\frac {(1+i)^{2}}{\operatorname {sl} ^{2}((1+i)z)}}} <p>we get</p> sl ′ z sl z = − ∑ n = 0 ∞ 2 4 n ( 2 − ( − 1 ) n 2 2 n ) H 4 n z 4 n − 1 ( 4 n ) ! , | z | < ϖ 2 . {\displaystyle {\frac {\operatorname {sl} 'z}{\operatorname {sl} z}}=-\sum _{n=0}^{\infty }2^{4n}(2-(-1)^{n}2^{2n})\mathrm {H} _{4n}{\frac {z^{4n-1}}{(4n)!}},\quad \left|z\right|<{\frac {\varpi }{\sqrt {2}}}.} <p>Hence</p> sl ′ z sl z = M ′ ( z ) M ( z ) − N ′ ( z ) N ( z ) , | z | < ϖ 2 . {\displaystyle {\frac {\operatorname {sl} 'z}{\operatorname {sl} z}}={\frac {M'(z)}{M(z)}}-{\frac {N'(z)}{N(z)}},\quad \left|z\right|<{\frac {\varpi }{\sqrt {2}}}.} <p>Therefore</p> sl z = C M ( z ) N ( z ) {\displaystyle \operatorname {sl} z=C{\frac {M(z)}{N(z)}}} <p>for some constant C {\displaystyle C} for | z | < ϖ / 2 {\displaystyle \left|z\right|<\varpi /{\sqrt {2}}} but this result holds for all z ∈ C {\displaystyle z\in \mathbb {C} } by analytic continuation. Using</p> lim z → 0 sl z z = 1 {\displaystyle \lim _{z\to 0}{\frac {\operatorname {sl} z}{z}}=1} <p>gives C = 1 {\displaystyle C=1} which completes the proof. ◼ {\displaystyle \blacksquare } </p><p>Proof by Liouville's theorem</p><p>Let</p> f ( z ) = M ( z ) N ( z ) = ( 1 + i ) M ( z ) 2 M ( ( 1 + i ) z ) , {\displaystyle f(z)={\frac {M(z)}{N(z)}}={\frac {(1+i)M(z)^{2}}{M((1+i)z)}},} <p>with patches at removable singularities.The shifting formulas</p> M ( z + 2 ϖ ) = e 2 π ϖ ( z + ϖ ) M ( z ) , M ( z + 2 ϖ i ) = e − 2 π ϖ ( i z − ϖ ) M ( z ) {\displaystyle M(z+2\varpi )=e^{2{\frac {\pi }{\varpi }}(z+\varpi )}M(z),\quad M(z+2\varpi i)=e^{-2{\frac {\pi }{\varpi }}(iz-\varpi )}M(z)} <p>imply that f {\displaystyle f} is an elliptic function with periods 2 ϖ {\displaystyle 2\varpi } and 2 ϖ i {\displaystyle 2\varpi i} , just as sl {\displaystyle \operatorname {sl} } .It follows that the function g {\displaystyle g} defined by</p> g ( z ) = sl z f ( z ) , {\displaystyle g(z)={\frac {\operatorname {sl} z}{f(z)}},} <p>when patched, is an elliptic function without poles. By <a href="/facts/Liouville%27s_theorem_(complex_analysis)/I97twMBQ">Liouville's theorem</a>, it is a constant. By using sl z = z + O ( z 5 ) {\displaystyle \operatorname {sl} z=z+\operatorname {O} (z^{5})} , M ( z ) = z + O ( z 5 ) {\displaystyle M(z)=z+\operatorname {O} (z^{5})} and N ( z ) = 1 + O ( z 4 ) {\displaystyle N(z)=1+\operatorname {O} (z^{4})} , this constant is 1 {\displaystyle 1} , which proves the theorem. ◼ {\displaystyle \blacksquare } </p></td></tr></tbody></table> <p>Gauss conjectured that ln N ( ϖ ) = π / 2 {\displaystyle \ln N(\varpi )=\pi /2} (this later turned out to be true) and commented that this “is most remarkable and a proof of this property promises the most serious increase in analysis”.<a class="footnote-ref" id="fnref:62" href="#fn:62"><sup>62</sup></a> Gauss expanded the products for M {\displaystyle M} and N {\displaystyle N} as infinite series (see below). He also discovered several identities involving the functions M {\displaystyle M} and N {\displaystyle N} , such as </p> N ( z ) = M ( ( 1 + i ) z ) ( 1 + i ) M ( z ) , z ∉ ϖ Z [ i ] {\displaystyle N(z)={\frac {M((1+i)z)}{(1+i)M(z)}},\quad z\notin \varpi \mathbb {Z} [i]} <p>and </p> N ( 2 z ) = M ( z ) 4 + N ( z ) 4 . {\displaystyle N(2z)=M(z)^{4}+N(z)^{4}.} <p>Thanks to a certain theorem<a class="footnote-ref" id="fnref:63" href="#fn:63"><sup>63</sup></a> on splitting limits, we are allowed to multiply out the infinite products and collect like powers of z {\displaystyle z} . Doing so gives the following power series expansions that are convergent everywhere in the complex plane:<a class="footnote-ref" id="fnref:64" href="#fn:64"><sup>64</sup></a><a class="footnote-ref" id="fnref:65" href="#fn:65"><sup>65</sup></a><a class="footnote-ref" id="fnref:66" href="#fn:66"><sup>66</sup></a><a class="footnote-ref" id="fnref:67" href="#fn:67"><sup>67</sup></a><a class="footnote-ref" id="fnref:68" href="#fn:68"><sup>68</sup></a> </p> M ( z ) = z − 2 z 5 5 ! − 36 z 9 9 ! + 552 z 13 13 ! + ⋯ , z ∈ C {\displaystyle M(z)=z-2{\frac {z^{5}}{5!}}-36{\frac {z^{9}}{9!}}+552{\frac {z^{13}}{13!}}+\cdots ,\quad z\in \mathbb {C} } N ( z ) = 1 + 2 z 4 4 ! − 4 z 8 8 ! + 408 z 12 12 ! + ⋯ , z ∈ C . {\displaystyle N(z)=1+2{\frac {z^{4}}{4!}}-4{\frac {z^{8}}{8!}}+408{\frac {z^{12}}{12!}}+\cdots ,\quad z\in \mathbb {C} .} <p>This can be contrasted with the power series of sl {\displaystyle \operatorname {sl} } which has only finite radius of convergence (because it is not entire). </p><p>We define S {\displaystyle S} and T {\displaystyle T} by </p> S ( z ) = N ( z 1 + i ) 2 − i M ( z 1 + i ) 2 , T ( z ) = S ( i z ) . {\displaystyle S(z)=N\left({\frac {z}{1+i}}\right)^{2}-iM\left({\frac {z}{1+i}}\right)^{2},\quad T(z)=S(iz).} <p>Then the lemniscate cosine can be written as </p> cl z = S ( z ) T ( z ) {\displaystyle \operatorname {cl} z={\frac {S(z)}{T(z)}}} <p>where<a class="footnote-ref" id="fnref:69" href="#fn:69"><sup>69</sup></a> </p> S ( z ) = 1 − z 2 2 ! − z 4 4 ! − 3 z 6 6 ! + 17 z 8 8 ! − 9 z 10 10 ! + 111 z 12 12 ! + ⋯ , z ∈ C {\displaystyle S(z)=1-{\frac {z^{2}}{2!}}-{\frac {z^{4}}{4!}}-3{\frac {z^{6}}{6!}}+17{\frac {z^{8}}{8!}}-9{\frac {z^{10}}{10!}}+111{\frac {z^{12}}{12!}}+\cdots ,\quad z\in \mathbb {C} } T ( z ) = 1 + z 2 2 ! − z 4 4 ! + 3 z 6 6 ! + 17 z 8 8 ! + 9 z 10 10 ! + 111 z 12 12 ! + ⋯ , z ∈ C . {\displaystyle T(z)=1+{\frac {z^{2}}{2!}}-{\frac {z^{4}}{4!}}+3{\frac {z^{6}}{6!}}+17{\frac {z^{8}}{8!}}+9{\frac {z^{10}}{10!}}+111{\frac {z^{12}}{12!}}+\cdots ,\quad z\in \mathbb {C} .} <p>Furthermore, the identities </p> M ( 2 z ) = 2 M ( z ) N ( z ) S ( z ) T ( z ) , {\displaystyle M(2z)=2M(z)N(z)S(z)T(z),} S ( 2 z ) = S ( z ) 4 − 2 M ( z ) 4 , {\displaystyle S(2z)=S(z)^{4}-2M(z)^{4},} T ( 2 z ) = T ( z ) 4 − 2 M ( z ) 4 {\displaystyle T(2z)=T(z)^{4}-2M(z)^{4}} <p>and the Pythagorean-like identities </p> M ( z ) 2 + S ( z ) 2 = N ( z ) 2 , {\displaystyle M(z)^{2}+S(z)^{2}=N(z)^{2},} M ( z ) 2 + N ( z ) 2 = T ( z ) 2 {\displaystyle M(z)^{2}+N(z)^{2}=T(z)^{2}} <p>hold for all z ∈ C {\displaystyle z\in \mathbb {C} } . </p><p>The quasi-addition formulas </p> M ( z + w ) M ( z − w ) = M ( z ) 2 N ( w ) 2 − N ( z ) 2 M ( w ) 2 , {\displaystyle M(z+w)M(z-w)=M(z)^{2}N(w)^{2}-N(z)^{2}M(w)^{2},} N ( z + w ) N ( z − w ) = N ( z ) 2 N ( w ) 2 + M ( z ) 2 M ( w ) 2 {\displaystyle N(z+w)N(z-w)=N(z)^{2}N(w)^{2}+M(z)^{2}M(w)^{2}} <p>(where z , w ∈ C {\displaystyle z,w\in \mathbb {C} } ) imply further multiplication formulas for M {\displaystyle M} and N {\displaystyle N} by recursion.<a class="footnote-ref" id="fnref:70" href="#fn:70"><sup>70</sup></a> </p><p>Gauss' M {\displaystyle M} and N {\displaystyle N} satisfy the following system of differential equations: </p> M ( z ) M ″ ( z ) = M ′ ( z ) 2 − N ( z ) 2 , {\displaystyle M(z)M''(z)=M'(z)^{2}-N(z)^{2},} N ( z ) N ″ ( z ) = N ′ ( z ) 2 + M ( z ) 2 {\displaystyle N(z)N''(z)=N'(z)^{2}+M(z)^{2}} <p>where z ∈ C {\displaystyle z\in \mathbb {C} } . Both M {\displaystyle M} and N {\displaystyle N} satisfy the differential equation<a class="footnote-ref" id="fnref:71" href="#fn:71"><sup>71</sup></a> </p> X ( z ) X ⁗ ( z ) = 4 X ′ ( z ) X ‴ ( z ) − 3 X ″ ( z ) 2 + 2 X ( z ) 2 , z ∈ C . {\displaystyle X(z)X''''(z)=4X'(z)X'''(z)-3X''(z)^{2}+2X(z)^{2},\quad z\in \mathbb {C} .} <p>The functions can be also expressed by integrals involving elliptic functions: </p> M ( z ) = z exp ( − ∫ 0 z ∫ 0 w ( 1 sl 2 v − 1 v 2 ) d v d w ) , {\displaystyle M(z)=z\exp \left(-\int _{0}^{z}\int _{0}^{w}\left({\frac {1}{\operatorname {sl} ^{2}v}}-{\frac {1}{v^{2}}}\right)\,\mathrm {d} v\,\mathrm {d} w\right),} N ( z ) = exp ( ∫ 0 z ∫ 0 w sl 2 v d v d w ) {\displaystyle N(z)=\exp \left(\int _{0}^{z}\int _{0}^{w}\operatorname {sl} ^{2}v\,\mathrm {d} v\,\mathrm {d} w\right)} <p>where the contours do not cross the poles; while the innermost integrals are path-independent, the outermost ones are path-dependent; however, the path dependence cancels out with the non-injectivity of the complex exponential function. </p><p>An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate elliptic functions § Methods of computation); the relation between M , N {\displaystyle M,N} and θ 1 , θ 3 {\displaystyle \theta _{1},\theta _{3}} is </p> M ( z ) = 2 − 1 / 4 e π z 2 / ( 2 ϖ 2 ) π ϖ θ 1 ( π z ϖ , e − π ) , {\displaystyle M(z)=2^{-1/4}e^{\pi z^{2}/(2\varpi ^{2})}{\sqrt {\frac {\pi }{\varpi }}}\theta _{1}\left({\frac {\pi z}{\varpi }},e^{-\pi }\right),} N ( z ) = 2 − 1 / 4 e π z 2 / ( 2 ϖ 2 ) π ϖ θ 3 ( π z ϖ , e − π ) {\displaystyle N(z)=2^{-1/4}e^{\pi z^{2}/(2\varpi ^{2})}{\sqrt {\frac {\pi }{\varpi }}}\theta _{3}\left({\frac {\pi z}{\varpi }},e^{-\pi }\right)} <p>where z ∈ C {\displaystyle z\in \mathbb {C} } . </p> <h2 id="relation-to-other-functions">Relation to other functions</h2> <h3>Relation to Weierstrass and Jacobi elliptic functions</h3> <p>The lemniscate functions are closely related to the <a href="/facts/Weierstrass_elliptic_function/GHRXDknk">Weierstrass elliptic function</a> ℘ ( z ; 1 , 0 ) {\displaystyle \wp (z;1,0)} (the "lemniscatic case"), with invariants <i>g</i>2 = 1 and <i>g</i>3 = 0. This lattice has fundamental periods ω 1 = 2 ϖ , {\displaystyle \omega _{1}={\sqrt {2}}\varpi ,} and ω 2 = i ω 1 {\displaystyle \omega _{2}=i\omega _{1}} . The associated constants of the Weierstrass function are e 1 = 1 2 , e 2 = 0 , e 3 = − 1 2 . {\displaystyle e_{1}={\tfrac {1}{2}},\ e_{2}=0,\ e_{3}=-{\tfrac {1}{2}}.} </p><p>The related case of a Weierstrass elliptic function with <i>g</i>2 = <i>a</i>, <i>g</i>3 = 0 may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: <i>a</i> > 0 and <i>a</i> < 0. The period <a href="/facts/Parallelogram/DRuktq60">parallelogram</a> is either a <a href="/facts/Square/JHJeMvmL">square</a> or a <a href="/facts/Rhombus/T4pU50GF">rhombus</a>. The Weierstrass elliptic function ℘ ( z ; − 1 , 0 ) {\displaystyle \wp (z;-1,0)} is called the "pseudolemniscatic case".<a class="footnote-ref" id="fnref:72" href="#fn:72"><sup>72</sup></a> </p><p>The square of the lemniscate sine can be represented as </p> sl 2 z = 1 ℘ ( z ; 4 , 0 ) = i 2 ℘ ( ( 1 − i ) z ; − 1 , 0 ) = − 2 ℘ ( 2 z + ( i − 1 ) ϖ 2 ; 1 , 0 ) {\displaystyle \operatorname {sl} ^{2}z={\frac {1}{\wp (z;4,0)}}={\frac {i}{2\wp ((1-i)z;-1,0)}}={-2\wp }{\left({\sqrt {2}}z+(i-1){\frac {\varpi }{\sqrt {2}}};1,0\right)}} <p>where the second and third argument of ℘ {\displaystyle \wp } denote the lattice invariants <i>g</i>2 and <i>g</i>3. The lemniscate sine is a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a> in the Weierstrass elliptic function and its derivative:<a class="footnote-ref" id="fnref:73" href="#fn:73"><sup>73</sup></a> </p> sl z = − 2 ℘ ( z ; − 1 , 0 ) ℘ ′ ( z ; − 1 , 0 ) . {\displaystyle \operatorname {sl} z=-2{\frac {\wp (z;-1,0)}{\wp '(z;-1,0)}}.} <p>The lemniscate functions can also be written in terms of <a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a>. The Jacobi elliptic functions sn {\displaystyle \operatorname {sn} } and cd {\displaystyle \operatorname {cd} } with positive real elliptic modulus have an "upright" rectangular lattice aligned with real and imaginary axes. Alternately, the functions sn {\displaystyle \operatorname {sn} } and cd {\displaystyle \operatorname {cd} } with modulus <i>i</i> (and sd {\displaystyle \operatorname {sd} } and cn {\displaystyle \operatorname {cn} } with modulus 1 / 2 {\displaystyle 1/{\sqrt {2}}} ) have a square period lattice rotated 1/8 turn.<a class="footnote-ref" id="fnref:74" href="#fn:74"><sup>74</sup></a><a class="footnote-ref" id="fnref:75" href="#fn:75"><sup>75</sup></a> </p> sl z = sn ( z ; i ) = sc ( z ; 2 ) = 1 2 sd ( 2 z ; 1 2 ) {\displaystyle \operatorname {sl} z=\operatorname {sn} (z;i)=\operatorname {sc} (z;{\sqrt {2}})={{\tfrac {1}{\sqrt {2}}}\operatorname {sd} }\left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right)} cl z = cd ( z ; i ) = dn ( z ; 2 ) = cn ( 2 z ; 1 2 ) {\displaystyle \operatorname {cl} z=\operatorname {cd} (z;i)=\operatorname {dn} (z;{\sqrt {2}})={\operatorname {cn} }\left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right)} <p>where the second arguments denote the elliptic modulus k {\displaystyle k} . </p><p>The functions sl ~ {\displaystyle {\tilde {\operatorname {sl} }}} and cl ~ {\displaystyle {\tilde {\operatorname {cl} }}} can also be expressed in terms of Jacobi elliptic functions: </p> sl ~ z = cd ( z ; i ) sd ( z ; i ) = dn ( z ; 2 ) sn ( z ; 2 ) = 1 2 cn ( 2 z ; 1 2 ) sn ( 2 z ; 1 2 ) , {\displaystyle {\tilde {\operatorname {sl} }}\,z=\operatorname {cd} (z;i)\operatorname {sd} (z;i)=\operatorname {dn} (z;{\sqrt {2}})\operatorname {sn} (z;{\sqrt {2}})={\tfrac {1}{\sqrt {2}}}\operatorname {cn} \left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right)\operatorname {sn} \left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right),} cl ~ z = cd ( z ; i ) nd ( z ; i ) = dn ( z ; 2 ) cn ( z ; 2 ) = cn ( 2 z ; 1 2 ) dn ( 2 z ; 1 2 ) . {\displaystyle {\tilde {\operatorname {cl} }}\,z=\operatorname {cd} (z;i)\operatorname {nd} (z;i)=\operatorname {dn} (z;{\sqrt {2}})\operatorname {cn} (z;{\sqrt {2}})=\operatorname {cn} \left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right)\operatorname {dn} \left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right).} <h3>Relation to the modular lambda function</h3> <p>The lemniscate sine can be used for the computation of values of the <a href="/facts/Modular_lambda_function/k30w0QWf">modular lambda function</a>: </p> ∏ k = 1 n sl ( 2 k − 1 2 n + 1 ϖ 2 ) = λ ( ( 2 n + 1 ) i ) 1 − λ ( ( 2 n + 1 ) i ) 8 {\displaystyle \prod _{k=1}^{n}\;{\operatorname {sl} }{\left({\frac {2k-1}{2n+1}}{\frac {\varpi }{2}}\right)}={\sqrt[{8}]{\frac {\lambda ((2n+1)i)}{1-\lambda ((2n+1)i)}}}} <p>For example: </p> sl ( 1 14 ϖ ) sl ( 3 14 ϖ ) sl ( 5 14 ϖ ) = λ ( 7 i ) 1 − λ ( 7 i ) 8 = tan ( 1 2 arccsc ( 1 2 8 7 + 21 + 1 2 7 + 1 ) ) = 2 2 + 7 + 21 + 8 7 + 2 14 + 6 7 + 455 + 172 7 sl ( 1 18 ϖ ) sl ( 3 18 ϖ ) sl ( 5 18 ϖ ) sl ( 7 18 ϖ ) = λ ( 9 i ) 1 − λ ( 9 i ) 8 = tan ( π 4 − arctan ( 2 2 3 − 2 3 − 2 2 − 3 3 + 3 − 1 12 4 ) ) {\displaystyle {\begin{aligned}&{\operatorname {sl} }{\bigl (}{\tfrac {1}{14}}\varpi {\bigr )}\,{\operatorname {sl} }{\bigl (}{\tfrac {3}{14}}\varpi {\bigr )}\,{\operatorname {sl} }{\bigl (}{\tfrac {5}{14}}\varpi {\bigr )}\\[7mu]&\quad {}={\sqrt[{8}]{\frac {\lambda (7i)}{1-\lambda (7i)}}}={\tan }{\Bigl (}{{\tfrac {1}{2}}\operatorname {arccsc} }{\Bigl (}{\tfrac {1}{2}}{\sqrt {8{\sqrt {7}}+21}}+{\tfrac {1}{2}}{\sqrt {7}}+1{\Bigr )}{\Bigr )}\\[7mu]&\quad {}={\frac {2}{2+{\sqrt {7}}+{\sqrt {21+8{\sqrt {7}}}}+{\sqrt {2{14+6{\sqrt {7}}+{\sqrt {455+172{\sqrt {7}}}}}}}}}\\[18mu]&{\operatorname {sl} }{\bigl (}{\tfrac {1}{18}}\varpi {\bigr )}\,{\operatorname {sl} }{\bigl (}{\tfrac {3}{18}}\varpi {\bigr )}\,{\operatorname {sl} }{\bigl (}{\tfrac {5}{18}}\varpi {\bigr )}\,{\operatorname {sl} }{\bigl (}{\tfrac {7}{18}}\varpi {\bigr )}\\[-3mu]&\quad {}={\sqrt[{8}]{\frac {\lambda (9i)}{1-\lambda (9i)}}}={\tan }{\Biggl (}{\frac {\pi }{4}}-{\arctan }{\Biggl (}{\frac {2{\sqrt[{3}]{2{\sqrt {3}}-2}}-2{\sqrt[{3}]{2-{\sqrt {3}}}}+{\sqrt {3}}-1}{\sqrt[{4}]{12}}}{\Biggr )}{\Biggr )}\end{aligned}}} <h2 id="inverse-functions">Inverse functions</h2> <p>The inverse function of the lemniscate sine is the lemniscate arcsine, defined as<a class="footnote-ref" id="fnref:76" href="#fn:76"><sup>76</sup></a> </p> arcsl x = ∫ 0 x d t 1 − t 4 . {\displaystyle \operatorname {arcsl} x=\int _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}.} <p>It can also be represented by the <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric function</a>: </p> arcsl x = x 2 F 1 ( 1 2 , 1 4 ; 5 4 ; x 4 ) {\displaystyle \operatorname {arcsl} x=x\,{}_{2}F_{1}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{4}};{\tfrac {5}{4}};x^{4}{\bigr )}} <p>which can be easily seen by using the <a href="/facts/Binomial_series/3JL2q2Ad">binomial series</a>. </p><p>The inverse function of the lemniscate cosine is the lemniscate arccosine. This function is defined by following expression: </p> arccl x = ∫ x 1 d t 1 − t 4 = 1 2 ϖ − arcsl x {\displaystyle \operatorname {arccl} x=\int _{x}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}={\tfrac {1}{2}}\varpi -\operatorname {arcsl} x} <p>For <i>x</i> in the interval − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} , sl arcsl x = x {\displaystyle \operatorname {sl} \operatorname {arcsl} x=x} and cl arccl x = x {\displaystyle \operatorname {cl} \operatorname {arccl} x=x} </p><p>For the halving of the lemniscate arc length these formulas are valid: </p> sl ( 1 2 arcsl x ) = sin ( 1 2 arcsin x ) sech ( 1 2 arsinh x ) sl ( 1 2 arcsl x ) 2 = tan ( 1 4 arcsin x 2 ) {\displaystyle {\begin{aligned}{\operatorname {sl} }{\bigl (}{\tfrac {1}{2}}\operatorname {arcsl} x{\bigr )}&={\sin }{\bigl (}{\tfrac {1}{2}}\arcsin x{\bigr )}\,{\operatorname {sech} }{\bigl (}{\tfrac {1}{2}}\operatorname {arsinh} x{\bigr )}\\{\operatorname {sl} }{\bigl (}{\tfrac {1}{2}}\operatorname {arcsl} x{\bigr )}^{2}&={\tan }{\bigl (}{\tfrac {1}{4}}\arcsin x^{2}{\bigr )}\end{aligned}}} <p>Furthermore there are the so called Hyperbolic lemniscate area functions: </p> aslh ( x ) = ∫ 0 x 1 y 4 + 1 d y = 1 2 F ( 2 arctan x ; 1 2 ) {\displaystyle \operatorname {aslh} (x)=\int _{0}^{x}{\frac {1}{\sqrt {y^{4}+1}}}\mathrm {d} y={\tfrac {1}{2}}F\left(2\arctan x;{\tfrac {1}{\sqrt {2}}}\right)} aclh ( x ) = ∫ x ∞ 1 y 4 + 1 d y = 1 2 F ( 2 arccot x ; 1 2 ) {\displaystyle \operatorname {aclh} (x)=\int _{x}^{\infty }{\frac {1}{\sqrt {y^{4}+1}}}\mathrm {d} y={\tfrac {1}{2}}F\left(2\operatorname {arccot} x;{\tfrac {1}{\sqrt {2}}}\right)} aclh ( x ) = ϖ 2 − aslh ( x ) {\displaystyle \operatorname {aclh} (x)={\frac {\varpi }{\sqrt {2}}}-\operatorname {aslh} (x)} aslh ( x ) = 2 arcsl ( x / 1 + x 4 + 1 ) {\displaystyle \operatorname {aslh} (x)={\sqrt {2}}\operatorname {arcsl} \left(x{\Big /}{\sqrt {\textstyle 1+{\sqrt {x^{4}+1}}}}\right)} arcsl ( x ) = 2 aslh ( x / 1 + 1 − x 4 ) {\displaystyle \operatorname {arcsl} (x)={\sqrt {2}}\operatorname {aslh} \left(x{\Big /}{\sqrt {\textstyle 1+{\sqrt {1-x^{4}}}}}\right)} <h3>Expression using elliptic integrals</h3> <p>The lemniscate arcsine and the lemniscate arccosine can also be expressed by the Legendre-Form: </p><p>These functions can be displayed directly by using the incomplete <a href="/facts/Elliptic_integral/q8EGYzQ3">elliptic integral</a> of the first kind: </p> arcsl x = 1 2 F ( arcsin 2 x 1 + x 2 ; 1 2 ) {\displaystyle \operatorname {arcsl} x={\frac {1}{\sqrt {2}}}F\left({\arcsin }{\frac {{\sqrt {2}}x}{\sqrt {1+x^{2}}}};{\frac {1}{\sqrt {2}}}\right)} arcsl x = 2 ( 2 − 1 ) F ( arcsin ( 2 + 1 ) x 1 + x 2 + 1 ; ( 2 − 1 ) 2 ) {\displaystyle \operatorname {arcsl} x=2({\sqrt {2}}-1)F\left({\arcsin }{\frac {({\sqrt {2}}+1)x}{{\sqrt {1+x^{2}}}+1}};({\sqrt {2}}-1)^{2}\right)} <p>The arc lengths of the lemniscate can also be expressed by only using the arc lengths of <a href="/facts/Ellipse/1wUDnGxY">ellipses</a> (calculated by elliptic integrals of the second kind): </p> arcsl x = 2 + 2 2 E ( arcsin ( 2 + 1 ) x 1 + x 2 + 1 ; ( 2 − 1 ) 2 ) − E ( arcsin 2 x 1 + x 2 ; 1 2 ) + x 1 − x 2 2 ( 1 + x 2 + 1 + x 2 ) {\displaystyle {\begin{aligned}\operatorname {arcsl} x={}&{\frac {2+{\sqrt {2}}}{2}}E\left({\arcsin }{\frac {({\sqrt {2}}+1)x}{{\sqrt {1+x^{2}}}+1}};({\sqrt {2}}-1)^{2}\right)\\[5mu]&\ \ -E\left({\arcsin }{\frac {{\sqrt {2}}x}{\sqrt {1+x^{2}}}};{\frac {1}{\sqrt {2}}}\right)+{\frac {x{\sqrt {1-x^{2}}}}{{\sqrt {2}}(1+x^{2}+{\sqrt {1+x^{2}}})}}\end{aligned}}} <p>The lemniscate arccosine has this expression: </p> arccl x = 1 2 F ( arccos x ; 1 2 ) {\displaystyle \operatorname {arccl} x={\frac {1}{\sqrt {2}}}F\left(\arccos x;{\frac {1}{\sqrt {2}}}\right)} <h3>Use in integration</h3> <p>The lemniscate arcsine can be used to integrate many functions. Here is a list of important integrals (the constants of integration are omitted): </p> ∫ 1 1 − x 4 d x = arcsl x {\displaystyle \int {\frac {1}{\sqrt {1-x^{4}}}}\,\mathrm {d} x=\operatorname {arcsl} x} ∫ 1 ( x 2 + 1 ) ( 2 x 2 + 1 ) d x = arcsl x x 2 + 1 {\displaystyle \int {\frac {1}{\sqrt {(x^{2}+1)(2x^{2}+1)}}}\,\mathrm {d} x={\operatorname {arcsl} }{\frac {x}{\sqrt {x^{2}+1}}}} ∫ 1 x 4 + 6 x 2 + 1 d x = arcsl 2 x x 4 + 6 x 2 + 1 + x 2 + 1 {\displaystyle \int {\frac {1}{\sqrt {x^{4}+6x^{2}+1}}}\,\mathrm {d} x={\operatorname {arcsl} }{\frac {{\sqrt {2}}x}{\sqrt {{\sqrt {x^{4}+6x^{2}+1}}+x^{2}+1}}}} ∫ 1 x 4 + 1 d x = 2 arcsl x x 4 + 1 + 1 {\displaystyle \int {\frac {1}{\sqrt {x^{4}+1}}}\,\mathrm {d} x={{\sqrt {2}}\operatorname {arcsl} }{\frac {x}{\sqrt {{\sqrt {x^{4}+1}}+1}}}} ∫ 1 ( 1 − x 4 ) 3 4 d x = 2 arcsl x 1 + 1 − x 4 {\displaystyle \int {\frac {1}{\sqrt[{4}]{(1-x^{4})^{3}}}}\,\mathrm {d} x={{\sqrt {2}}\operatorname {arcsl} }{\frac {x}{\sqrt {1+{\sqrt {1-x^{4}}}}}}} ∫ 1 ( x 4 + 1 ) 3 4 d x = arcsl x x 4 + 1 4 {\displaystyle \int {\frac {1}{\sqrt[{4}]{(x^{4}+1)^{3}}}}\,\mathrm {d} x={\operatorname {arcsl} }{\frac {x}{\sqrt[{4}]{x^{4}+1}}}} ∫ 1 ( 1 − x 2 ) 3 4 d x = 2 arcsl x 1 + 1 − x 2 {\displaystyle \int {\frac {1}{\sqrt[{4}]{(1-x^{2})^{3}}}}\,\mathrm {d} x={2\operatorname {arcsl} }{\frac {x}{1+{\sqrt {1-x^{2}}}}}} ∫ 1 ( x 2 + 1 ) 3 4 d x = 2 arcsl x x 2 + 1 + 1 {\displaystyle \int {\frac {1}{\sqrt[{4}]{(x^{2}+1)^{3}}}}\,\mathrm {d} x={2\operatorname {arcsl} }{\frac {x}{{\sqrt {x^{2}+1}}+1}}} ∫ 1 ( a x 2 + b x + c ) 3 4 d x = 2 2 4 a 2 c − a b 2 4 arcsl 2 a x + b 4 a ( a x 2 + b x + c ) + 4 a c − b 2 {\displaystyle \int {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}\,\mathrm {d} x={{\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} }{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}} ∫ sech x d x = 2 arcsl tanh 1 2 x {\displaystyle \int {\sqrt {\operatorname {sech} x}}\,\mathrm {d} x={2\operatorname {arcsl} }\tanh {\tfrac {1}{2}}x} ∫ sec x d x = 2 arcsl tan 1 2 x {\displaystyle \int {\sqrt {\sec x}}\,\mathrm {d} x={2\operatorname {arcsl} }\tan {\tfrac {1}{2}}x} <h2 id="hyperbolic-lemniscate-functions">Hyperbolic lemniscate functions</h2> <h3>Fundamental information</h3> <p>For convenience, let σ = 2 ϖ {\displaystyle \sigma ={\sqrt {2}}\varpi } . σ {\displaystyle \sigma } is the "squircular" analog of π {\displaystyle \pi } (see below). The decimal expansion of σ {\displaystyle \sigma } (i.e. 3.7081 … {\displaystyle 3.7081\ldots } <a class="footnote-ref" id="fnref:77" href="#fn:77"><sup>77</sup></a>) appears in entry 34e of chapter 11 of Ramanujan's second notebook.<a class="footnote-ref" id="fnref:78" href="#fn:78"><sup>78</sup></a> </p><p>The hyperbolic lemniscate sine (slh) and cosine (clh) can be defined as inverses of elliptic integrals as follows: </p> z = ∗ ∫ 0 slh z d t 1 + t 4 = ∫ clh z ∞ d t 1 + t 4 {\displaystyle z\mathrel {\overset {*}{=}} \int _{0}^{\operatorname {slh} z}{\frac {\mathrm {d} t}{\sqrt {1+t^{4}}}}=\int _{\operatorname {clh} z}^{\infty }{\frac {\mathrm {d} t}{\sqrt {1+t^{4}}}}} <p>where in ( ∗ ) {\displaystyle (*)} , z {\displaystyle z} is in the square with corners { σ / 2 , σ i / 2 , − σ / 2 , − σ i / 2 } {\displaystyle \{\sigma /2,\sigma i/2,-\sigma /2,-\sigma i/2\}} . Beyond that square, the functions can be analytically continued to meromorphic functions in the whole complex plane. </p><p>The complete integral has the value: </p> ∫ 0 ∞ d t t 4 + 1 = 1 4 B ( 1 4 , 1 4 ) = σ 2 = 1.85407 46773 01371 … {\displaystyle \int _{0}^{\infty }{\frac {\mathrm {d} t}{\sqrt {t^{4}+1}}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{4}}{\bigr )}={\frac {\sigma }{2}}=1.85407\;46773\;01371\ldots } <p>Therefore, the two defined functions have following relation to each other: </p> slh z = clh ( σ 2 − z ) {\displaystyle \operatorname {slh} z={\operatorname {clh} }{{\Bigl (}{\frac {\sigma }{2}}-z{\Bigr )}}} <p>The product of hyperbolic lemniscate sine and hyperbolic lemniscate cosine is equal to one: </p> slh z clh z = 1 {\displaystyle \operatorname {slh} z\,\operatorname {clh} z=1} <p>The functions slh {\displaystyle \operatorname {slh} } and clh {\displaystyle \operatorname {clh} } have a square period lattice with fundamental periods { σ , σ i } {\displaystyle \{\sigma ,\sigma i\}} . </p><p>The hyperbolic lemniscate functions can be expressed in terms of lemniscate sine and lemniscate cosine: </p> slh ( 2 z ) = ( 1 + cl 2 z ) sl z 2 cl z {\displaystyle \operatorname {slh} {\bigl (}{\sqrt {2}}z{\bigr )}={\frac {(1+\operatorname {cl} ^{2}z)\operatorname {sl} z}{{\sqrt {2}}\operatorname {cl} z}}} clh ( 2 z ) = ( 1 + sl 2 z ) cl z 2 sl z {\displaystyle \operatorname {clh} {\bigl (}{\sqrt {2}}z{\bigr )}={\frac {(1+\operatorname {sl} ^{2}z)\operatorname {cl} z}{{\sqrt {2}}\operatorname {sl} z}}} <p>But there is also a relation to the <a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a> with the elliptic modulus one by square root of two: </p> slh z = sn ( z ; 1 / 2 ) cd ( z ; 1 / 2 ) {\displaystyle \operatorname {slh} z={\frac {\operatorname {sn} (z;1/{\sqrt {2}})}{\operatorname {cd} (z;1/{\sqrt {2}})}}} clh z = cd ( z ; 1 / 2 ) sn ( z ; 1 / 2 ) {\displaystyle \operatorname {clh} z={\frac {\operatorname {cd} (z;1/{\sqrt {2}})}{\operatorname {sn} (z;1/{\sqrt {2}})}}} <p>The hyperbolic lemniscate sine has following imaginary relation to the lemniscate sine: </p> slh z = 1 − i 2 sl ( 1 + i 2 z ) = sl ( − 1 4 z ) − 1 4 {\displaystyle \operatorname {slh} z={\frac {1-i}{\sqrt {2}}}\operatorname {sl} \left({\frac {1+i}{\sqrt {2}}}z\right)={\frac {\operatorname {sl} \left({\sqrt[{4}]{-1}}z\right)}{\sqrt[{4}]{-1}}}} <p>This is analogous to the relationship between hyperbolic and trigonometric sine: </p> sinh z = − i sin ( i z ) = sin ( − 1 2 z ) − 1 2 {\displaystyle \sinh z=-i\sin(iz)={\frac {\sin \left({\sqrt[{2}]{-1}}z\right)}{\sqrt[{2}]{-1}}}} <h3>Relation to quartic Fermat curve</h3> <h4>Hyperbolic Lemniscate Tangent and Cotangent</h4> <p>This image shows the standardized superelliptic Fermat squircle curve of the fourth degree: </p> <p>In a quartic <a href="/facts/Fermat_curve/31LFyJ5p">Fermat curve</a> x 4 + y 4 = 1 {\displaystyle x^{4}+y^{4}=1} (sometimes called a <a href="/facts/Squircle/KECRKzRt">squircle</a>) the hyperbolic lemniscate sine and cosine are analogous to the tangent and cotangent functions in a unit circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} (the quadratic Fermat curve). If the origin and a point on the curve are connected to each other by a line <i>L</i>, the hyperbolic lemniscate sine of twice the enclosed area between this line and the x-axis is the y-coordinate of the intersection of <i>L</i> with the line x = 1 {\displaystyle x=1} .<a class="footnote-ref" id="fnref:79" href="#fn:79"><sup>79</sup></a> Just as π {\displaystyle \pi } is the area enclosed by the circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} , the area enclosed by the squircle x 4 + y 4 = 1 {\displaystyle x^{4}+y^{4}=1} is σ {\displaystyle \sigma } . Moreover, </p> M ( 1 , 1 / 2 ) = π σ {\displaystyle M(1,1/{\sqrt {2}})={\frac {\pi }{\sigma }}} <p>where M {\displaystyle M} is the <a href="/facts/Arithmetic%E2%80%93geometric_mean/8PRN9UB4">arithmetic–geometric mean</a>. </p><p>The hyperbolic lemniscate sine satisfies the argument addition identity: </p> slh ( a + b ) = slh a slh ′ b + slh b slh ′ a 1 − slh 2 a slh 2 b {\displaystyle \operatorname {slh} (a+b)={\frac {\operatorname {slh} a\operatorname {slh} 'b+\operatorname {slh} b\operatorname {slh} 'a}{1-\operatorname {slh} ^{2}a\,\operatorname {slh} ^{2}b}}} <p>When u {\displaystyle u} is real, the derivative and the original antiderivative of slh {\displaystyle \operatorname {slh} } and clh {\displaystyle \operatorname {clh} } can be expressed in this way: </p> <table><tbody><tr><td><p> d d u slh ( u ) = 1 + slh ( u ) 4 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} u}}\operatorname {slh} (u)={\sqrt {1+\operatorname {slh} (u)^{4}}}} </p><p> d d u clh ( u ) = − 1 + clh ( u ) 4 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} u}}\operatorname {clh} (u)=-{\sqrt {1+\operatorname {clh} (u)^{4}}}} </p><p> d d u 1 2 arsinh [ slh ( u ) 2 ] = slh ( u ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} u}}\,{\frac {1}{2}}\operatorname {arsinh} {\bigl [}\operatorname {slh} (u)^{2}{\bigr ]}=\operatorname {slh} (u)} </p><p> d d u − 1 2 arsinh [ clh ( u ) 2 ] = clh ( u ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} u}}-\,{\frac {1}{2}}\operatorname {arsinh} {\bigl [}\operatorname {clh} (u)^{2}{\bigr ]}=\operatorname {clh} (u)} </p></td></tr></tbody></table> <p>There are also the Hyperbolic lemniscate tangent and the Hyperbolic lemniscate coangent als further functions: </p><p>The functions tlh and ctlh fulfill the identities described in the differential equation mentioned: </p> tlh ( 2 u ) = sin 4 ( 2 u ) = sl ( u ) cl 2 u + 1 sl 2 u + cl 2 u {\displaystyle {\text{tlh}}({\sqrt {2}}\,u)=\sin _{4}({\sqrt {2}}\,u)=\operatorname {sl} (u){\sqrt {\frac {\operatorname {cl} ^{2}u+1}{\operatorname {sl} ^{2}u+\operatorname {cl} ^{2}u}}}} ctlh ( 2 u ) = cos 4 ( 2 u ) = cl ( u ) sl 2 u + 1 sl 2 u + cl 2 u {\displaystyle {\text{ctlh}}({\sqrt {2}}\,u)=\cos _{4}({\sqrt {2}}\,u)=\operatorname {cl} (u){\sqrt {\frac {\operatorname {sl} ^{2}u+1}{\operatorname {sl} ^{2}u+\operatorname {cl} ^{2}u}}}} <p>The functional designation sl stands for the lemniscatic sine and the designation cl stands for the lemniscatic cosine. In addition, those relations to the <a href="/facts/Jacobi_elliptic_function/BLk6Rxew">Jacobi elliptic functions</a> are valid: </p> tlh ( u ) = sn ( u ; 1 2 2 ) cd ( u ; 1 2 2 ) 4 + sn ( u ; 1 2 2 ) 4 4 {\displaystyle {\text{tlh}}(u)={\frac {{\text{sn}}(u;{\tfrac {1}{2}}{\sqrt {2}})}{\sqrt[{4}]{{\text{cd}}(u;{\tfrac {1}{2}}{\sqrt {2}})^{4}+{\text{sn}}(u;{\tfrac {1}{2}}{\sqrt {2}})^{4}}}}} ctlh ( u ) = cd ( u ; 1 2 2 ) cd ( u ; 1 2 2 ) 4 + sn ( u ; 1 2 2 ) 4 4 {\displaystyle {\text{ctlh}}(u)={\frac {{\text{cd}}(u;{\tfrac {1}{2}}{\sqrt {2}})}{\sqrt[{4}]{{\text{cd}}(u;{\tfrac {1}{2}}{\sqrt {2}})^{4}+{\text{sn}}(u;{\tfrac {1}{2}}{\sqrt {2}})^{4}}}}} <p>When u {\displaystyle u} is real, the derivative and quarter period integral of tlh {\displaystyle \operatorname {tlh} } and ctlh {\displaystyle \operatorname {ctlh} } can be expressed in this way: </p> <table><tbody><tr><td><p> d d u tlh ( u ) = ctlh ( u ) 3 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} u}}\operatorname {tlh} (u)=\operatorname {ctlh} (u)^{3}} </p><p> d d u ctlh ( u ) = − tlh ( u ) 3 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} u}}\operatorname {ctlh} (u)=-\operatorname {tlh} (u)^{3}} </p><p> ∫ 0 ϖ / 2 tlh ( u ) d u = ϖ 2 {\displaystyle \int _{0}^{\varpi /{\sqrt {2}}}\operatorname {tlh} (u)\,\mathrm {d} u={\frac {\varpi }{2}}} </p><p> ∫ 0 ϖ / 2 ctlh ( u ) d u = ϖ 2 {\displaystyle \int _{0}^{\varpi /{\sqrt {2}}}\operatorname {ctlh} (u)\,\mathrm {d} u={\frac {\varpi }{2}}} </p></td></tr></tbody></table> <h4>Derivation of the Hyperbolic Lemniscate functions</h4> <p>The horizontal and vertical coordinates of this superellipse are dependent on twice the enclosed area w = 2A, so the following conditions must be met: </p> x ( w ) 4 + y ( w ) 4 = 1 {\displaystyle x(w)^{4}+y(w)^{4}=1} d d w x ( w ) = − y ( w ) 3 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} w}}x(w)=-y(w)^{3}} d d w y ( w ) = x ( w ) 3 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} w}}y(w)=x(w)^{3}} x ( w = 0 ) = 1 {\displaystyle x(w=0)=1} y ( w = 0 ) = 0 {\displaystyle y(w=0)=0} <p>The solutions to this system of equations are as follows: </p> x ( w ) = cl ( 1 2 2 w ) [ sl ( 1 2 2 w ) 2 + 1 ] 1 / 2 [ sl ( 1 2 2 w ) 2 + cl ( 1 2 2 w ) 2 ] − 1 / 2 {\displaystyle x(w)=\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}w)[\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}+1]^{1/2}[\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}+\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}]^{-1/2}} y ( w ) = sl ( 1 2 2 w ) [ cl ( 1 2 2 w ) 2 + 1 ] 1 / 2 [ sl ( 1 2 2 w ) 2 + cl ( 1 2 2 w ) 2 ] − 1 / 2 {\displaystyle y(w)=\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}w)[\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}+1]^{1/2}[\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}+\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}]^{-1/2}} <p>The following therefore applies to the quotient: </p> y ( w ) x ( w ) = sl ( 1 2 2 w ) [ cl ( 1 2 2 w ) 2 + 1 ] 1 / 2 cl ( 1 2 2 w ) [ sl ( 1 2 2 w ) 2 + 1 ] 1 / 2 = slh ( w ) {\displaystyle {\frac {y(w)}{x(w)}}={\frac {\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}w)[\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}+1]^{1/2}}{\operatorname {cl} ({\tfrac {1}{2}}{\sqrt {2}}w)[\operatorname {sl} ({\tfrac {1}{2}}{\sqrt {2}}w)^{2}+1]^{1/2}}}=\operatorname {slh} (w)} <p>The functions x(w) and y(w) are called cotangent hyperbolic lemniscatus and hyperbolic tangent. </p> x ( w ) = ctlh ( w ) {\displaystyle x(w)={\text{ctlh}}(w)} y ( w ) = tlh ( w ) {\displaystyle y(w)={\text{tlh}}(w)} <p>The sketch also shows the fact that the derivation of the Areasinus hyperbolic lemniscatus function is equal to the reciprocal of the square root of the successor of the fourth power function. </p> <h4>First proof: comparison with the derivative of the arctangent</h4> <p>There is a black diagonal on the sketch shown on the right. The length of the segment that runs perpendicularly from the intersection of this black diagonal with the red vertical axis to the point (1|0) should be called s. And the length of the section of the black diagonal from the coordinate origin point to the point of intersection of this diagonal with the cyan curved line of the superellipse has the following value depending on the slh value: </p> D ( s ) = ( 1 s 4 + 1 4 ) 2 + ( s s 4 + 1 4 ) 2 = s 2 + 1 s 4 + 1 4 {\displaystyle D(s)={\sqrt {{\biggl (}{\frac {1}{\sqrt[{4}]{s^{4}+1}}}{\biggr )}^{2}+{\biggl (}{\frac {s}{\sqrt[{4}]{s^{4}+1}}}{\biggr )}^{2}}}={\frac {\sqrt {s^{2}+1}}{\sqrt[{4}]{s^{4}+1}}}} <p>This connection is described by the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a>. </p><p>An analogous unit circle results in the arctangent of the circle trigonometric with the described area allocation. </p><p>The following derivation applies to this: </p> d d s arctan ( s ) = 1 s 2 + 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} s}}\arctan(s)={\frac {1}{s^{2}+1}}} <p>To determine the derivation of the areasinus lemniscatus hyperbolicus, the comparison of the infinitesimally small triangular areas for the same diagonal in the superellipse and the unit circle is set up below. Because the summation of the infinitesimally small triangular areas describes the area dimensions. In the case of the superellipse in the picture, half of the area concerned is shown in green. Because of the quadratic ratio of the areas to the lengths of triangles with the same infinitesimally small angle at the origin of the coordinates, the following formula applies: </p> d d s aslh ( s ) = [ d d s arctan ( s ) ] D ( s ) 2 = 1 s 2 + 1 D ( s ) 2 = 1 s 2 + 1 ( s 2 + 1 s 4 + 1 4 ) 2 = 1 s 4 + 1 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} s}}{\text{aslh}}(s)={\biggl [}{\frac {\mathrm {d} }{\mathrm {d} s}}\arctan(s){\biggr ]}D(s)^{2}={\frac {1}{s^{2}+1}}D(s)^{2}={\frac {1}{s^{2}+1}}{\biggl (}{\frac {\sqrt {s^{2}+1}}{\sqrt[{4}]{s^{4}+1}}}{\biggr )}^{2}={\frac {1}{\sqrt {s^{4}+1}}}} <h4>Second proof: integral formation and area subtraction</h4> <p>In the picture shown, the area tangent lemniscatus hyperbolicus assigns the height of the intersection of the diagonal and the curved line to twice the green area. The green area itself is created as the difference integral of the superellipse function from zero to the relevant height value minus the area of the adjacent triangle: </p> atlh ( v ) = 2 ( ∫ 0 v 1 − w 4 4 d w ) − v 1 − v 4 4 {\displaystyle {\text{atlh}}(v)=2{\biggl (}\int _{0}^{v}{\sqrt[{4}]{1-w^{4}}}\mathrm {d} w{\biggr )}-v{\sqrt[{4}]{1-v^{4}}}} d d v atlh ( v ) = 2 1 − v 4 4 − ( d d v v 1 − v 4 4 ) = 1 ( 1 − v 4 ) 3 / 4 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} v}}{\text{atlh}}(v)=2{\sqrt[{4}]{1-v^{4}}}-{\biggl (}{\frac {\mathrm {d} }{\mathrm {d} v}}v{\sqrt[{4}]{1-v^{4}}}{\biggr )}={\frac {1}{(1-v^{4})^{3/4}}}} <p>The following transformation applies: </p> aslh ( x ) = atlh ( x x 4 + 1 4 ) {\displaystyle {\text{aslh}}(x)={\text{atlh}}{\biggl (}{\frac {x}{\sqrt[{4}]{x^{4}+1}}}{\biggr )}} <p>And so, according to the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a>, this derivation holds: </p> d d x aslh ( x ) = d d x atlh ( x x 4 + 1 4 ) = ( d d x x x 4 + 1 4 ) [ 1 − ( x x 4 + 1 4 ) 4 ] − 3 / 4 = {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\text{aslh}}(x)={\frac {\mathrm {d} }{\mathrm {d} x}}{\text{atlh}}{\biggl (}{\frac {x}{\sqrt[{4}]{x^{4}+1}}}{\biggr )}={\biggl (}{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {x}{\sqrt[{4}]{x^{4}+1}}}{\biggr )}{\biggl [}1-{\biggl (}{\frac {x}{\sqrt[{4}]{x^{4}+1}}}{\biggr )}^{4}{\biggr ]}^{-3/4}=} = 1 ( x 4 + 1 ) 5 / 4 [ 1 − ( x x 4 + 1 4 ) 4 ] − 3 / 4 = 1 ( x 4 + 1 ) 5 / 4 ( 1 x 4 + 1 ) − 3 / 4 = 1 x 4 + 1 {\displaystyle ={\frac {1}{(x^{4}+1)^{5/4}}}{\biggl [}1-{\biggl (}{\frac {x}{\sqrt[{4}]{x^{4}+1}}}{\biggr )}^{4}{\biggr ]}^{-3/4}={\frac {1}{(x^{4}+1)^{5/4}}}{\biggl (}{\frac {1}{x^{4}+1}}{\biggr )}^{-3/4}={\frac {1}{\sqrt {x^{4}+1}}}} <h3>Specific values</h3> <p>This list shows the values of the Hyperbolic Lemniscate Sine accurately. Recall that, </p> ∫ 0 ∞ d t t 4 + 1 = 1 4 B ( 1 4 , 1 4 ) = ϖ 2 = σ 2 = 1.85407 … {\displaystyle \int _{0}^{\infty }{\frac {\operatorname {d} t}{\sqrt {t^{4}+1}}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{4}}{\bigr )}={\frac {\varpi }{\sqrt {2}}}={\frac {\sigma }{2}}=1.85407\ldots } <p>whereas 1 2 B ( 1 2 , 1 2 ) = π 2 , {\displaystyle {\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}={\tfrac {\pi }{2}},} so the values below such as slh ( ϖ 2 2 ) = slh ( σ 4 ) = 1 {\displaystyle {\operatorname {slh} }{\bigl (}{\tfrac {\varpi }{2{\sqrt {2}}}}{\bigr )}={\operatorname {slh} }{\bigl (}{\tfrac {\sigma }{4}}{\bigr )}=1} are analogous to the trigonometric sin ( π 2 ) = 1 {\displaystyle {\sin }{\bigl (}{\tfrac {\pi }{2}}{\bigr )}=1} . </p> slh ( ϖ 2 2 ) = 1 {\displaystyle \operatorname {slh} \,\left({\frac {\varpi }{2{\sqrt {2}}}}\right)=1} slh ( ϖ 3 2 ) = 1 3 4 2 3 − 3 4 {\displaystyle \operatorname {slh} \,\left({\frac {\varpi }{3{\sqrt {2}}}}\right)={\frac {1}{\sqrt[{4}]{3}}}{\sqrt[{4}]{2{\sqrt {3}}-3}}} slh ( 2 ϖ 3 2 ) = 2 3 + 3 4 {\displaystyle \operatorname {slh} \,\left({\frac {2\varpi }{3{\sqrt {2}}}}\right)={\sqrt[{4}]{2{\sqrt {3}}+3}}} slh ( ϖ 4 2 ) = 1 2 4 ( 2 + 1 − 1 ) {\displaystyle \operatorname {slh} \,\left({\frac {\varpi }{4{\sqrt {2}}}}\right)={\frac {1}{\sqrt[{4}]{2}}}({\sqrt {{\sqrt {2}}+1}}-1)} slh ( 3 ϖ 4 2 ) = 1 2 4 ( 2 + 1 + 1 ) {\displaystyle \operatorname {slh} \,\left({\frac {3\varpi }{4{\sqrt {2}}}}\right)={\frac {1}{\sqrt[{4}]{2}}}({\sqrt {{\sqrt {2}}+1}}+1)} slh ( ϖ 5 2 ) = 1 8 4 5 − 1 20 4 − 5 + 1 = 2 5 − 2 4 sin ( 1 20 π ) sin ( 3 20 π ) {\displaystyle \operatorname {slh} \,\left({\frac {\varpi }{5{\sqrt {2}}}}\right)={\frac {1}{\sqrt[{4}]{8}}}{\sqrt {{\sqrt {5}}-1}}{\sqrt {{\sqrt[{4}]{20}}-{\sqrt {{\sqrt {5}}+1}}}}=2{\sqrt[{4}]{{\sqrt {5}}-2}}{\sqrt {\sin({\tfrac {1}{20}}\pi )\sin({\tfrac {3}{20}}\pi )}}} slh ( 2 ϖ 5 2 ) = 1 2 2 4 ( 5 + 1 ) 20 4 − 5 + 1 = 2 5 + 2 4 sin ( 1 20 π ) sin ( 3 20 π ) {\displaystyle \operatorname {slh} \,\left({\frac {2\varpi }{5{\sqrt {2}}}}\right)={\frac {1}{2{\sqrt[{4}]{2}}}}({\sqrt {5}}+1){\sqrt {{\sqrt[{4}]{20}}-{\sqrt {{\sqrt {5}}+1}}}}=2{\sqrt[{4}]{{\sqrt {5}}+2}}{\sqrt {\sin({\tfrac {1}{20}}\pi )\sin({\tfrac {3}{20}}\pi )}}} slh ( 3 ϖ 5 2 ) = 1 8 4 5 − 1 20 4 + 5 + 1 = 2 5 − 2 4 cos ( 1 20 π ) cos ( 3 20 π ) {\displaystyle \operatorname {slh} \,\left({\frac {3\varpi }{5{\sqrt {2}}}}\right)={\frac {1}{\sqrt[{4}]{8}}}{\sqrt {{\sqrt {5}}-1}}{\sqrt {{\sqrt[{4}]{20}}+{\sqrt {{\sqrt {5}}+1}}}}=2{\sqrt[{4}]{{\sqrt {5}}-2}}{\sqrt {\cos({\tfrac {1}{20}}\pi )\cos({\tfrac {3}{20}}\pi )}}} slh ( 4 ϖ 5 2 ) = 1 2 2 4 ( 5 + 1 ) 20 4 + 5 + 1 = 2 5 + 2 4 cos ( 1 20 π ) cos ( 3 20 π ) {\displaystyle \operatorname {slh} \,\left({\frac {4\varpi }{5{\sqrt {2}}}}\right)={\frac {1}{2{\sqrt[{4}]{2}}}}({\sqrt {5}}+1){\sqrt {{\sqrt[{4}]{20}}+{\sqrt {{\sqrt {5}}+1}}}}=2{\sqrt[{4}]{{\sqrt {5}}+2}}{\sqrt {\cos({\tfrac {1}{20}}\pi )\cos({\tfrac {3}{20}}\pi )}}} slh ( ϖ 6 2 ) = 1 2 ( 2 3 + 3 + 1 ) ( 1 − 2 3 − 3 4 ) {\displaystyle \operatorname {slh} \,\left({\frac {\varpi }{6{\sqrt {2}}}}\right)={\frac {1}{2}}({\sqrt {2{\sqrt {3}}+3}}+1)(1-{\sqrt[{4}]{2{\sqrt {3}}-3}})} slh ( 5 ϖ 6 2 ) = 1 2 ( 2 3 + 3 + 1 ) ( 1 + 2 3 − 3 4 ) {\displaystyle \operatorname {slh} \,\left({\frac {5\varpi }{6{\sqrt {2}}}}\right)={\frac {1}{2}}({\sqrt {2{\sqrt {3}}+3}}+1)(1+{\sqrt[{4}]{2{\sqrt {3}}-3}})} <p>That table shows the most important values of the Hyperbolic Lemniscate Tangent and Cotangent functions: </p> <table><tbody><tr><th> z {\displaystyle z} </th><th> clh z {\displaystyle \operatorname {clh} z} </th><th> slh z {\displaystyle \operatorname {slh} z} </th><th> ctlh z = cos 4 z {\displaystyle \operatorname {ctlh} z=\cos _{4}z} </th><th> tlh z = sin 4 z {\displaystyle \operatorname {tlh} z=\sin _{4}z} </th></tr><tr><td> 0 {\displaystyle 0} </td><td> ∞ {\displaystyle \infty } </td><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 1 4 σ {\displaystyle {\tfrac {1}{4}}\sigma } </td><td> 1 {\displaystyle 1} </td><td> 1 {\displaystyle 1} </td><td> 1 / 2 4 {\displaystyle 1{\big /}{\sqrt[{4}]{2}}} </td><td> 1 / 2 4 {\displaystyle 1{\big /}{\sqrt[{4}]{2}}} </td></tr><tr><td> 1 2 σ {\displaystyle {\tfrac {1}{2}}\sigma } </td><td> 0 {\displaystyle 0} </td><td> ∞ {\displaystyle \infty } </td><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td></tr><tr><td> 3 4 σ {\displaystyle {\tfrac {3}{4}}\sigma } </td><td> − 1 {\displaystyle -1} </td><td> − 1 {\displaystyle -1} </td><td> − 1 / 2 4 {\displaystyle -1{\big /}{\sqrt[{4}]{2}}} </td><td> 1 / 2 4 {\displaystyle 1{\big /}{\sqrt[{4}]{2}}} </td></tr><tr><td> σ {\displaystyle \sigma } </td><td> ∞ {\displaystyle \infty } </td><td> 0 {\displaystyle 0} </td><td> − 1 {\displaystyle -1} </td><td> 0 {\displaystyle 0} </td></tr></tbody></table> <h3>Combination and halving theorems</h3> <p>Given the <i>hyperbolic lemniscate tangent</i> ( tlh {\displaystyle \operatorname {tlh} } ) and <i>hyperbolic lemniscate cotangent</i> ( ctlh {\displaystyle \operatorname {ctlh} } ). Recall the <i>hyperbolic lemniscate area functions</i> from the section on inverse functions, </p> aslh ( x ) = ∫ 0 x 1 y 4 + 1 d y {\displaystyle \operatorname {aslh} (x)=\int _{0}^{x}{\frac {1}{\sqrt {y^{4}+1}}}\mathrm {d} y} aclh ( x ) = ∫ x ∞ 1 y 4 + 1 d y {\displaystyle \operatorname {aclh} (x)=\int _{x}^{\infty }{\frac {1}{\sqrt {y^{4}+1}}}\mathrm {d} y} <p>Then the following identities can be established, </p> tlh [ aslh ( x ) ] = ctlh [ aclh ( x ) ] = x x 4 + 1 4 {\displaystyle {\text{tlh}}{\bigl [}{\text{aslh}}(x){\bigr ]}={\text{ctlh}}{\bigl [}{\text{aclh}}(x){\bigr ]}={\frac {x}{\sqrt[{4}]{x^{4}+1}}}} ctlh [ aslh ( x ) ] = tlh [ aclh ( x ) ] = 1 x 4 + 1 4 {\displaystyle {\text{ctlh}}{\bigl [}{\text{aslh}}(x){\bigr ]}={\text{tlh}}{\bigl [}{\text{aclh}}(x){\bigr ]}={\frac {1}{\sqrt[{4}]{x^{4}+1}}}} <p>hence the 4th power of tlh {\displaystyle \operatorname {tlh} } and ctlh {\displaystyle \operatorname {ctlh} } for these arguments is equal to one, </p> tlh [ aslh ( x ) ] 4 + ctlh [ aslh ( x ) ] 4 = 1 {\displaystyle {\text{tlh}}{\bigl [}{\text{aslh}}(x){\bigr ]}^{4}+{\text{ctlh}}{\bigl [}{\text{aslh}}(x){\bigr ]}^{4}=1} tlh [ aclh ( x ) ] 4 + ctlh [ aclh ( x ) ] 4 = 1 {\displaystyle {\text{tlh}}{\bigl [}{\text{aclh}}(x){\bigr ]}^{4}+{\text{ctlh}}{\bigl [}{\text{aclh}}(x){\bigr ]}^{4}=1} <p>so a 4th power version of the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a>. The bisection theorem of the hyperbolic sinus lemniscatus reads as follows: </p> slh [ 1 2 aslh ( x ) ] = 2 x x 2 + 1 + x 4 + 1 + x 4 + 1 − x 2 + 1 {\displaystyle {\text{slh}}{\bigl [}{\tfrac {1}{2}}{\text{aslh}}(x){\bigr ]}={\frac {{\sqrt {2}}x}{{\sqrt {x^{2}+1+{\sqrt {x^{4}+1}}}}+{\sqrt {{\sqrt {x^{4}+1}}-x^{2}+1}}}}} <p>This formula can be revealed as a combination of the following two formulas: </p> a s l h ( x ) = 2 arcsl [ x ( x 4 + 1 + 1 ) − 1 / 2 ] {\displaystyle \mathrm {aslh} (x)={\sqrt {2}}\,{\text{arcsl}}{\bigl [}x({\sqrt {x^{4}+1}}+1)^{-1/2}{\bigr ]}} arcsl ( x ) = 2 aslh ( 2 x 1 + x 2 + 1 − x 2 ) {\displaystyle {\text{arcsl}}(x)={\sqrt {2}}\,{\text{aslh}}{\bigl (}{\frac {{\sqrt {2}}x}{{\sqrt {1+x^{2}}}+{\sqrt {1-x^{2}}}}}{\bigr )}} <p>In addition, the following formulas are valid for all real values x ∈ R {\displaystyle x\in \mathbb {R} } : </p> slh [ 1 2 aclh ( x ) ] = x 4 + 1 + x 2 − 2 x x 4 + 1 + x 2 = ( x 4 + 1 − x 2 + 1 ) − 1 / 2 ( x 4 + 1 + 1 − x ) {\displaystyle {\text{slh}}{\bigl [}{\tfrac {1}{2}}{\text{aclh}}(x){\bigr ]}={\sqrt {{\sqrt {x^{4}+1}}+x^{2}-{\sqrt {2}}x{\sqrt {{\sqrt {x^{4}+1}}+x^{2}}}}}={\bigl (}{\sqrt {x^{4}+1}}-x^{2}+1{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {x^{4}+1}}+1}}-x{\bigr )}} clh [ 1 2 aclh ( x ) ] = x 4 + 1 + x 2 + 2 x x 4 + 1 + x 2 = ( x 4 + 1 − x 2 + 1 ) − 1 / 2 ( x 4 + 1 + 1 + x ) {\displaystyle {\text{clh}}{\bigl [}{\tfrac {1}{2}}{\text{aclh}}(x){\bigr ]}={\sqrt {{\sqrt {x^{4}+1}}+x^{2}+{\sqrt {2}}x{\sqrt {{\sqrt {x^{4}+1}}+x^{2}}}}}={\bigl (}{\sqrt {x^{4}+1}}-x^{2}+1{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {x^{4}+1}}+1}}+x{\bigr )}} <p>These identities follow from the last-mentioned formula: </p> tlh [ 1 2 aclh ( x ) ] 2 = 1 2 2 − 2 2 x x 4 + 1 − x 2 = ( 2 x 2 + 2 + 2 x 4 + 1 ) − 1 / 2 ( x 4 + 1 + 1 − x ) {\displaystyle {\text{tlh}}[{\tfrac {1}{2}}{\text{aclh}}(x)]^{2}={\tfrac {1}{2}}{\sqrt {2-2{\sqrt {2}}\,x{\sqrt {{\sqrt {x^{4}+1}}-x^{2}}}}}={\bigl (}2x^{2}+2+2{\sqrt {x^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {x^{4}+1}}+1}}-x{\bigr )}} ctlh [ 1 2 aclh ( x ) ] 2 = 1 2 2 + 2 2 x x 4 + 1 − x 2 = ( 2 x 2 + 2 + 2 x 4 + 1 ) − 1 / 2 ( x 4 + 1 + 1 + x ) {\displaystyle {\text{ctlh}}[{\tfrac {1}{2}}{\text{aclh}}(x)]^{2}={\tfrac {1}{2}}{\sqrt {2+2{\sqrt {2}}\,x{\sqrt {{\sqrt {x^{4}+1}}-x^{2}}}}}={\bigl (}2x^{2}+2+2{\sqrt {x^{4}+1}}{\bigr )}^{-1/2}{\bigl (}{\sqrt {{\sqrt {x^{4}+1}}+1}}+x{\bigr )}} <p>Hence, their 4th powers again equal one, </p> tlh [ 1 2 aclh ( x ) ] 4 + ctlh [ 1 2 aclh ( x ) ] 4 = 1 {\displaystyle {\text{tlh}}{\bigl [}{\tfrac {1}{2}}{\text{aclh}}(x){\bigr ]}^{4}+{\text{ctlh}}{\bigl [}{\tfrac {1}{2}}{\text{aclh}}(x){\bigr ]}^{4}=1} <p>The following formulas for the lemniscatic sine and lemniscatic cosine are closely related: </p> sl [ 1 2 2 aclh ( x ) ] = cl [ 1 2 2 aslh ( x ) ] = x 4 + 1 − x 2 {\displaystyle {\text{sl}}[{\tfrac {1}{2}}{\sqrt {2}}\,{\text{aclh}}(x)]={\text{cl}}[{\tfrac {1}{2}}{\sqrt {2}}\,{\text{aslh}}(x)]={\sqrt {{\sqrt {x^{4}+1}}-x^{2}}}} sl [ 1 2 2 aslh ( x ) ] = cl [ 1 2 2 aclh ( x ) ] = x ( x 4 + 1 + 1 ) − 1 / 2 {\displaystyle {\text{sl}}[{\tfrac {1}{2}}{\sqrt {2}}\,{\text{aslh}}(x)]={\text{cl}}[{\tfrac {1}{2}}{\sqrt {2}}\,{\text{aclh}}(x)]=x{\bigl (}{\sqrt {x^{4}+1}}+1{\bigr )}^{-1/2}} <h3>Coordinate Transformations</h3> <p>Analogous to the determination of the improper integral in the <a href="/facts/Normal_distribution/UapjjPyQ">Gaussian bell curve function</a>, the coordinate transformation of a <a href="/facts/Cylinder_(geometry)/0wQPPr3k">general cylinder</a> can be used to calculate the integral from 0 to the positive infinity in the function f ( x ) = exp ( − x 4 ) {\displaystyle f(x)=\exp(-x^{4})} integrated in relation to x. In the following, the proofs of both integrals are given in a parallel way of displaying. </p><p>This is the <a href="/facts/Polar_coordinates/HBko4YoN">cylindrical coordinate transformation</a> in the Gaussian bell curve function: </p> [ ∫ 0 ∞ exp ( − x 2 ) d x ] 2 = ∫ 0 ∞ ∫ 0 ∞ exp ( − y 2 − z 2 ) d y d z = {\displaystyle {\biggl [}\int _{0}^{\infty }\exp(-x^{2})\,\mathrm {d} x{\biggr ]}^{2}=\int _{0}^{\infty }\int _{0}^{\infty }\exp(-y^{2}-z^{2})\,\mathrm {d} y\,\mathrm {d} z=} = ∫ 0 π / 2 ∫ 0 ∞ det [ ∂ / ∂ r r cos ( ϕ ) ∂ / ∂ ϕ r cos ( ϕ ) ∂ / ∂ r r sin ( ϕ ) ∂ / ∂ ϕ r sin ( ϕ ) ] exp { − [ r cos ( ϕ ) ] 2 − [ r sin ( ϕ ) ] 2 } d r d ϕ = {\displaystyle =\int _{0}^{\pi /2}\int _{0}^{\infty }\det {\begin{bmatrix}\partial /\partial r\,\,r\cos(\phi )&\partial /\partial \phi \,\,r\cos(\phi )\\\partial /\partial r\,\,r\sin(\phi )&\partial /\partial \phi \,\,r\sin(\phi )\end{bmatrix}}\exp {\bigl \{}-{\bigl [}r\cos(\phi ){\bigr ]}^{2}-{\bigl [}r\sin(\phi ){\bigr ]}^{2}{\bigr \}}\,\mathrm {d} r\,\mathrm {d} \phi =} = ∫ 0 π / 2 ∫ 0 ∞ r exp ( − r 2 ) d r d ϕ = ∫ 0 π / 2 1 2 d ϕ = π 4 {\displaystyle =\int _{0}^{\pi /2}\int _{0}^{\infty }r\exp(-r^{2})\,\mathrm {d} r\,\mathrm {d} \phi =\int _{0}^{\pi /2}{\frac {1}{2}}\,\mathrm {d} \phi ={\frac {\pi }{4}}} <p>And this is the analogous coordinate transformation for the lemniscatory case: </p> [ ∫ 0 ∞ exp ( − x 4 ) d x ] 2 = ∫ 0 ∞ ∫ 0 ∞ exp ( − y 4 − z 4 ) d y d z = {\displaystyle {\biggl [}\int _{0}^{\infty }\exp(-x^{4})\,\mathrm {d} x{\biggr ]}^{2}=\int _{0}^{\infty }\int _{0}^{\infty }\exp(-y^{4}-z^{4})\,\mathrm {d} y\,\mathrm {d} z=} = ∫ 0 ϖ / 2 ∫ 0 ∞ det [ ∂ / ∂ r r ctlh ( ϕ ) ∂ / ∂ ϕ r ctlh ( ϕ ) ∂ / ∂ r r tlh ( ϕ ) ∂ / ∂ ϕ r tlh ( ϕ ) ] exp { − [ r ctlh ( ϕ ) ] 4 − [ r tlh ( ϕ ) ] 4 } d r d ϕ = {\displaystyle =\int _{0}^{\varpi /{\sqrt {2}}}\int _{0}^{\infty }\det {\begin{bmatrix}\partial /\partial r\,\,r\,{\text{ctlh}}(\phi )&\partial /\partial \phi \,\,r\,{\text{ctlh}}(\phi )\\\partial /\partial r\,\,r\,{\text{tlh}}(\phi )&\partial /\partial \phi \,\,r\,{\text{tlh}}(\phi )\end{bmatrix}}\exp {\bigl \{}-{\bigl [}r\,{\text{ctlh}}(\phi ){\bigr ]}^{4}-{\bigl [}r\,{\text{tlh}}(\phi ){\bigr ]}^{4}{\bigr \}}\,\mathrm {d} r\,\mathrm {d} \phi =} = ∫ 0 ϖ / 2 ∫ 0 ∞ r exp ( − r 4 ) d r d ϕ = ∫ 0 ϖ / 2 π 4 d ϕ = ϖ π 4 2 {\displaystyle =\int _{0}^{\varpi /{\sqrt {2}}}\int _{0}^{\infty }r\exp(-r^{4})\,\mathrm {d} r\,\mathrm {d} \phi =\int _{0}^{\varpi /{\sqrt {2}}}{\frac {\sqrt {\pi }}{4}}\,\mathrm {d} \phi ={\frac {\varpi {\sqrt {\pi }}}{4{\sqrt {2}}}}} <p>In the last line of this elliptically analogous equation chain there is again the original Gauss bell curve integrated with the square function as the inner substitution according to the <a href="/facts/Chain_rule/aMLYxP0x">Chain rule</a> of infinitesimal analytics (analysis). </p><p>In both cases, the determinant of the <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobi matrix</a> is multiplied to the original function in the integration domain. </p><p>The resulting new functions in the integration area are then integrated according to the new parameters. </p> <h2 id="number-theory">Number theory</h2> <p>In <a href="/facts/Algebraic_number_theory/LPzDT5CA">algebraic number theory</a>, every finite <a href="/facts/Abelian_extension/vIdW4Hnv">abelian extension</a> of the <a href="/facts/Gaussian_rational/X0jN1MnI">Gaussian rationals</a> Q ( i ) {\displaystyle \mathbb {Q} (i)} is a <a href="/facts/Field_extension/L7yIDtyK">subfield</a> of Q ( i , ω n ) {\displaystyle \mathbb {Q} (i,\omega _{n})} for some positive integer n {\displaystyle n} .<a class="footnote-ref" id="fnref:80" href="#fn:80"><sup>80</sup></a><a class="footnote-ref" id="fnref:81" href="#fn:81"><sup>81</sup></a> This is analogous to the <a href="/facts/Kronecker%E2%80%93Weber_theorem/U3pq1GAq">Kronecker–Weber theorem</a> for the rational numbers Q {\displaystyle \mathbb {Q} } which is based on division of the circle – in particular, every finite abelian extension of Q {\displaystyle \mathbb {Q} } is a subfield of Q ( ζ n ) {\displaystyle \mathbb {Q} (\zeta _{n})} for some positive integer n {\displaystyle n} . Both are special cases of Kronecker's Jugendtraum, which became <a href="/facts/Hilbert%27s_twelfth_problem/9II5Tu28">Hilbert's twelfth problem</a>. </p><p>The <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> Q ( i , sl ( ϖ / n ) ) {\displaystyle \mathbb {Q} (i,\operatorname {sl} (\varpi /n))} (for positive odd n {\displaystyle n} ) is the extension of Q ( i ) {\displaystyle \mathbb {Q} (i)} generated by the x {\displaystyle x} - and y {\displaystyle y} -coordinates of the ( 1 + i ) n {\displaystyle (1+i)n} -<a href="/facts/Torsion_(algebra)/uNRnwtLz">torsion points</a> on the <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a> y 2 = 4 x 3 + x {\displaystyle y^{2}=4x^{3}+x} .<a class="footnote-ref" id="fnref:82" href="#fn:82"><sup>82</sup></a> </p> <h3>Hurwitz numbers</h3> <p>The <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli numbers</a> B n {\displaystyle \mathrm {B} _{n}} can be defined by </p> B n = lim z → 0 d n d z n z e z − 1 , n ≥ 0 {\displaystyle \mathrm {B} _{n}=\lim _{z\to 0}{\frac {\mathrm {d} ^{n}}{\mathrm {d} z^{n}}}{\frac {z}{e^{z}-1}},\quad n\geq 0} <p>and appear in </p> ∑ k ∈ Z ∖ { 0 } 1 k 2 n = ( − 1 ) n − 1 B 2 n ( 2 π ) 2 n ( 2 n ) ! = 2 ζ ( 2 n ) , n ≥ 1 {\displaystyle \sum _{k\in \mathbb {Z} \setminus \{0\}}{\frac {1}{k^{2n}}}=(-1)^{n-1}\mathrm {B} _{2n}{\frac {(2\pi )^{2n}}{(2n)!}}=2\zeta (2n),\quad n\geq 1} <p>where ζ {\displaystyle \zeta } is the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>. </p><p>The Hurwitz numbers H n , {\displaystyle \mathrm {H} _{n},} named after <a href="/facts/Adolf_Hurwitz/urNgHh58">Adolf Hurwitz</a>, are the "lemniscate analogs" of the Bernoulli numbers. They can be defined by<a class="footnote-ref" id="fnref:83" href="#fn:83"><sup>83</sup></a><a class="footnote-ref" id="fnref:84" href="#fn:84"><sup>84</sup></a> </p> H n = − lim z → 0 d n d z n z ζ ( z ; 1 / 4 , 0 ) , n ≥ 0 {\displaystyle \mathrm {H} _{n}=-\lim _{z\to 0}{\frac {\mathrm {d} ^{n}}{\mathrm {d} z^{n}}}z\zeta (z;1/4,0),\quad n\geq 0} <p>where ζ ( ⋅ ; 1 / 4 , 0 ) {\displaystyle \zeta (\cdot ;1/4,0)} is the <a href="/facts/Weierstrass_functions/88h4u2hI">Weierstrass zeta function</a> with lattice invariants 1 / 4 {\displaystyle 1/4} and 0 {\displaystyle 0} . They appear in </p> ∑ z ∈ Z [ i ] ∖ { 0 } 1 z 4 n = H 4 n ( 2 ϖ ) 4 n ( 4 n ) ! = G 4 n ( i ) , n ≥ 1 {\displaystyle \sum _{z\in \mathbb {Z} [i]\setminus \{0\}}{\frac {1}{z^{4n}}}=\mathrm {H} _{4n}{\frac {(2\varpi )^{4n}}{(4n)!}}=G_{4n}(i),\quad n\geq 1} <p>where Z [ i ] {\displaystyle \mathbb {Z} [i]} are the <a href="/facts/Gaussian_integers/zj82AhtL">Gaussian integers</a> and G 4 n {\displaystyle G_{4n}} are the <a href="/facts/Eisenstein_series/23vgJLFl">Eisenstein series</a> of weight 4 n {\displaystyle 4n} , and in </p> ∑ n = 1 ∞ n k e 2 π n − 1 = { 1 24 − 1 8 π if k = 1 B k + 1 2 k + 2 if k ≡ 1 ( m o d 4 ) and k ≥ 5 B k + 1 2 k + 2 + H k + 1 2 k + 2 ( ϖ π ) k + 1 if k ≡ 3 ( m o d 4 ) and k ≥ 3. {\displaystyle \displaystyle {\begin{array}{ll}\displaystyle \sum _{n=1}^{\infty }{\dfrac {n^{k}}{e^{2\pi n}-1}}={\begin{cases}{\dfrac {1}{24}}-{\dfrac {1}{8\pi }}&{\text{if}}\ k=1\\{\dfrac {\mathrm {B} _{k+1}}{2k+2}}&{\text{if}}\ k\equiv 1\,(\mathrm {mod} \,4)\ {\text{and}}\ k\geq 5\\{\dfrac {\mathrm {B} _{k+1}}{2k+2}}+{\dfrac {\mathrm {H} _{k+1}}{2k+2}}\left({\dfrac {\varpi }{\pi }}\right)^{k+1}&{\text{if}}\ k\equiv 3\,(\mathrm {mod} \,4)\ {\text{and}}\ k\geq 3.\\\end{cases}}\end{array}}} <p>The Hurwitz numbers can also be determined as follows: H 4 = 1 / 10 {\displaystyle \mathrm {H} _{4}=1/10} , </p> H 4 n = 3 ( 2 n − 3 ) ( 16 n 2 − 1 ) ∑ k = 1 n − 1 ( 4 n 4 k ) ( 4 k − 1 ) ( 4 ( n − k ) − 1 ) H 4 k H 4 ( n − k ) , n ≥ 2 {\displaystyle \mathrm {H} _{4n}={\frac {3}{(2n-3)(16n^{2}-1)}}\sum _{k=1}^{n-1}{\binom {4n}{4k}}(4k-1)(4(n-k)-1)\mathrm {H} _{4k}\mathrm {H} _{4(n-k)},\quad n\geq 2} <p>and H n = 0 {\displaystyle \mathrm {H} _{n}=0} if n {\displaystyle n} is not a multiple of 4 {\displaystyle 4} .<a class="footnote-ref" id="fnref:85" href="#fn:85"><sup>85</sup></a> This yields<a class="footnote-ref" id="fnref:86" href="#fn:86"><sup>86</sup></a> </p> H 8 = 3 10 , H 12 = 567 130 , H 16 = 43 659 170 , … {\displaystyle \mathrm {H} _{8}={\frac {3}{10}},\,\mathrm {H} _{12}={\frac {567}{130}},\,\mathrm {H} _{16}={\frac {43\,659}{170}},\,\ldots } <p>Also<a class="footnote-ref" id="fnref:87" href="#fn:87"><sup>87</sup></a> </p> denom H 4 n = ∏ ( p − 1 ) | 4 n p {\displaystyle \operatorname {denom} \mathrm {H} _{4n}=\prod _{(p-1)|4n}p} <p>where p ∈ P {\displaystyle p\in \mathbb {P} } such that p ≢ 3 ( mod 4 ) , {\displaystyle p\not \equiv 3\,({\text{mod}}\,4),} just as </p> denom B 2 n = ∏ ( p − 1 ) | 2 n p {\displaystyle \operatorname {denom} \mathrm {B} _{2n}=\prod _{(p-1)|2n}p} <p>where p ∈ P {\displaystyle p\in \mathbb {P} } (by the <a href="/facts/Von_Staudt%E2%80%93Clausen_theorem/wH5xBFrV">von Staudt–Clausen theorem</a>). </p><p>In fact, the von Staudt–Clausen theorem determines the <a href="/facts/Fractional_part/lSU6Ap1q">fractional part</a> of the Bernoulli numbers: </p> B 2 n + ∑ ( p − 1 ) | 2 n 1 p ∈ Z , n ≥ 1 {\displaystyle \mathrm {B} _{2n}+\sum _{(p-1)|2n}{\frac {1}{p}}\in \mathbb {Z} ,\quad n\geq 1} <p>(sequence A000146 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>) where p {\displaystyle p} is any prime, and an analogous theorem holds for the Hurwitz numbers: suppose that a ∈ Z {\displaystyle a\in \mathbb {Z} } is odd, b ∈ Z {\displaystyle b\in \mathbb {Z} } is even, p {\displaystyle p} is a prime such that p ≡ 1 ( m o d 4 ) {\displaystyle p\equiv 1\,(\mathrm {mod} \,4)} , p = a 2 + b 2 {\displaystyle p=a^{2}+b^{2}} (see <a href="/facts/Fermat%27s_theorem_on_sums_of_two_squares/9y6knjq9">Fermat's theorem on sums of two squares</a>) and a ≡ b + 1 ( m o d 4 ) {\displaystyle a\equiv b+1\,(\mathrm {mod} \,4)} . Then for any given p {\displaystyle p} , 2 a = ν ( p ) {\displaystyle 2a=\nu (p)} is uniquely determined; equivalently ν ( p ) = p − N p {\displaystyle \nu (p)=p-{\mathcal {N}}_{p}} where N p {\displaystyle {\mathcal {N}}_{p}} is the number of solutions of the congruence X 3 − X ≡ Y 2 ( mod p ) {\displaystyle X^{3}-X\equiv Y^{2}\,(\operatorname {mod} p)} in variables X , Y {\displaystyle X,Y} that are non-negative integers.<a class="footnote-ref" id="fnref:88" href="#fn:88"><sup>88</sup></a> The Hurwitz theorem then determines the fractional part of the Hurwitz numbers:<a class="footnote-ref" id="fnref:89" href="#fn:89"><sup>89</sup></a> </p> H 4 n − 1 2 − ∑ ( p − 1 ) | 4 n ν ( p ) 4 n / ( p − 1 ) p = def G n ∈ Z , n ≥ 1. {\displaystyle \mathrm {H} _{4n}-{\frac {1}{2}}-\sum _{(p-1)|4n}{\frac {\nu (p)^{4n/(p-1)}}{p}}\mathrel {\overset {\text{def}}{=}} \mathrm {G} _{n}\in \mathbb {Z} ,\quad n\geq 1.} <p>The sequence of the integers G n {\displaystyle \mathrm {G} _{n}} starts with 0 , − 1 , 5 , 253 , … . {\displaystyle 0,-1,5,253,\ldots .} <a class="footnote-ref" id="fnref:90" href="#fn:90"><sup>90</sup></a> </p><p>Let n ≥ 2 {\displaystyle n\geq 2} . If 4 n + 1 {\displaystyle 4n+1} is a prime, then G n ≡ 1 ( m o d 4 ) {\displaystyle \mathrm {G} _{n}\equiv 1\,(\mathrm {mod} \,4)} . If 4 n + 1 {\displaystyle 4n+1} is not a prime, then G n ≡ 3 ( m o d 4 ) {\displaystyle \mathrm {G} _{n}\equiv 3\,(\mathrm {mod} \,4)} .<a class="footnote-ref" id="fnref:91" href="#fn:91"><sup>91</sup></a> </p><p>Some authors instead define the Hurwitz numbers as H n ′ = H 4 n {\displaystyle \mathrm {H} _{n}'=\mathrm {H} _{4n}} . </p> <h4>Appearances in Laurent series</h4> <p>The Hurwitz numbers appear in several <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a> expansions related to the lemniscate functions:<a class="footnote-ref" id="fnref:92" href="#fn:92"><sup>92</sup></a> </p> sl 2 z = ∑ n = 1 ∞ 2 4 n ( 1 − ( − 1 ) n 2 2 n ) H 4 n 4 n z 4 n − 2 ( 4 n − 2 ) ! , | z | < ϖ 2 sl ′ z sl z = 1 z − ∑ n = 1 ∞ 2 4 n ( 2 − ( − 1 ) n 2 2 n ) H 4 n 4 n z 4 n − 1 ( 4 n − 1 ) ! , | z | < ϖ 2 1 sl z = 1 z − ∑ n = 1 ∞ 2 2 n ( ( − 1 ) n 2 − 2 2 n ) H 4 n 4 n z 4 n − 1 ( 4 n − 1 ) ! , | z | < ϖ 1 sl 2 z = 1 z 2 + ∑ n = 1 ∞ 2 4 n H 4 n 4 n z 4 n − 2 ( 4 n − 2 ) ! , | z | < ϖ {\displaystyle {\begin{aligned}\operatorname {sl} ^{2}z&=\sum _{n=1}^{\infty }{\frac {2^{4n}(1-(-1)^{n}2^{2n})\mathrm {H} _{4n}}{4n}}{\frac {z^{4n-2}}{(4n-2)!}},\quad \left|z\right|<{\frac {\varpi }{\sqrt {2}}}\\{\frac {\operatorname {sl} 'z}{\operatorname {sl} {z}}}&={\frac {1}{z}}-\sum _{n=1}^{\infty }{\frac {2^{4n}(2-(-1)^{n}2^{2n})\mathrm {H} _{4n}}{4n}}{\frac {z^{4n-1}}{(4n-1)!}},\quad \left|z\right|<{\frac {\varpi }{\sqrt {2}}}\\{\frac {1}{\operatorname {sl} z}}&={\frac {1}{z}}-\sum _{n=1}^{\infty }{\frac {2^{2n}((-1)^{n}2-2^{2n})\mathrm {H} _{4n}}{4n}}{\frac {z^{4n-1}}{(4n-1)!}},\quad \left|z\right|<\varpi \\{\frac {1}{\operatorname {sl} ^{2}z}}&={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }{\frac {2^{4n}\mathrm {H} _{4n}}{4n}}{\frac {z^{4n-2}}{(4n-2)!}},\quad \left|z\right|<\varpi \end{aligned}}} <p>Analogously, in terms of the Bernoulli numbers: </p> 1 sinh 2 z = 1 z 2 − ∑ n = 1 ∞ 2 2 n B 2 n 2 n z 2 n − 2 ( 2 n − 2 ) ! , | z | < π . {\displaystyle {\frac {1}{\sinh ^{2}z}}={\frac {1}{z^{2}}}-\sum _{n=1}^{\infty }{\frac {2^{2n}\mathrm {B} _{2n}}{2n}}{\frac {z^{2n-2}}{(2n-2)!}},\quad \left|z\right|<\pi .} <h3>A quartic analog of the Legendre symbol</h3> <p>Let p {\displaystyle p} be a prime such that p ≡ 1 ( mod 4 ) {\displaystyle p\equiv 1\,({\text{mod}}\,4)} . A quartic residue (mod p {\displaystyle p} ) is any number congruent to the fourth power of an integer. Define ( a p ) 4 {\displaystyle \left({\tfrac {a}{p}}\right)_{4}} to be 1 {\displaystyle 1} if a {\displaystyle a} is a quartic residue (mod p {\displaystyle p} ) and define it to be − 1 {\displaystyle -1} if a {\displaystyle a} is not a quartic residue (mod p {\displaystyle p} ). </p><p>If a {\displaystyle a} and p {\displaystyle p} are coprime, then there exist numbers p ′ ∈ Z [ i ] {\displaystyle p'\in \mathbb {Z} [i]} (see<a class="footnote-ref" id="fnref:93" href="#fn:93"><sup>93</sup></a> for these numbers) such that<a class="footnote-ref" id="fnref:94" href="#fn:94"><sup>94</sup></a> </p> ( a p ) 4 = ∏ p ′ sl ( 2 ϖ a p ′ / p ) sl ( 2 ϖ p ′ / p ) . {\displaystyle \left({\frac {a}{p}}\right)_{4}=\prod _{p'}{\frac {\operatorname {sl} (2\varpi ap'/p)}{\operatorname {sl} (2\varpi p'/p)}}.} <p>This theorem is analogous to </p> ( a p ) = ∏ n = 1 p − 1 2 sin ( 2 π a n / p ) sin ( 2 π n / p ) {\displaystyle \left({\frac {a}{p}}\right)=\prod _{n=1}^{\frac {p-1}{2}}{\frac {\sin(2\pi an/p)}{\sin(2\pi n/p)}}} <p>where ( ⋅ ⋅ ) {\displaystyle \left({\tfrac {\cdot }{\cdot }}\right)} is the <a href="/facts/Legendre_symbol/MwHrJ7as">Legendre symbol</a>. </p> <h2 id="world-map-projections">World map projections</h2> <p>The <a href="/facts/Peirce_quincuncial_projection/t9pdVUsy">Peirce quincuncial projection</a>, designed by <a href="/facts/Charles_Sanders_Peirce/z76dNqhM">Charles Sanders Peirce</a> of the <a href="/facts/United_States_Coast_and_Geodetic_Survey/8XhGwogo">US Coast Survey</a> in the 1870s, is a world <a href="/facts/Map_projection/XYxklzIu">map projection</a> based on the inverse lemniscate sine of <a href="/facts/Stereographic_projection/oVyMrWqR">stereographically projected</a> points (treated as complex numbers).<a class="footnote-ref" id="fnref:95" href="#fn:95"><sup>95</sup></a> </p><p>When lines of constant real or imaginary part are projected onto the complex plane via the hyperbolic lemniscate sine, and thence stereographically projected onto the sphere (see <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a>), the resulting curves are <a href="/facts/Spherical_conic/fCqzrPmU">spherical conics</a>, the spherical analog of planar <a href="/facts/Ellipse/1wUDnGxY">ellipses</a> and <a href="/facts/Hyperbola/6R6ER0mY">hyperbolas</a>.<a class="footnote-ref" id="fnref:96" href="#fn:96"><sup>96</sup></a> Thus the lemniscate functions (and more generally, the <a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a>) provide a parametrization for spherical conics. </p><p>A conformal map projection from the globe onto the 6 square faces of a <a href="/facts/Cube/65lUaAqN">cube</a> can also be defined using the lemniscate functions.<a class="footnote-ref" id="fnref:97" href="#fn:97"><sup>97</sup></a> Because many <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a> can be effectively solved by conformal mapping, this map from sphere to cube is convenient for <a href="/facts/Atmospheric_model/j7epJxSw">atmospheric modeling</a>.<a class="footnote-ref" id="fnref:98" href="#fn:98"><sup>98</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Elliptic_function/mF03oI04">Elliptic function</a> <ul><li><a href="/facts/Abel_elliptic_functions/JqolraWM">Abel elliptic functions</a></li> <li><a href="/facts/Dixon_elliptic_functions/PIjCB2qS">Dixon elliptic functions</a></li> <li><a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a></li> <li><a href="/facts/Weierstrass_elliptic_function/GHRXDknk">Weierstrass elliptic function</a></li></ul></li> <li><a href="/facts/Elliptic_Gauss_sum/wYmamsQR">Elliptic Gauss sum</a></li> <li><a href="/facts/Lemniscate_constant/jzkcBdBL">Lemniscate constant</a></li> <li><a href="/facts/Peirce_quincuncial_projection/t9pdVUsy">Peirce quincuncial projection</a></li> <li><a href="/facts/Schwarz%E2%80%93Christoffel_mapping/vzHbRDDG">Schwarz–Christoffel mapping</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Matt_Parker/Sj4pTjWw">Parker, Matt</a> (2021). <a href="https://www.youtube.com/watch?v=gjtTcyWL0NA">"What is the area of a Squircle?"</a>. <i>Stand-up Maths</i>. YouTube. <a href="https://ghostarchive.org/varchive/youtube/20211219/gjtTcyWL0NA">Archived</a> from the original on 2021-12-19.</li></ul> <ul><li>Ramalingam, Muthu Veerappan (2023). <a href="https://www.youtube.com/watch?v=mAzIE5OkqWE">"Bernoulli Lemniscate and the Squircle | | {\displaystyle ||} A remarkable Geometric fun fact!!?"</a>. <i>Act of Learning</i>. YouTube. <a href="https://ghostarchive.org/varchive/mAzIE5OkqWE">Archived</a> from the original on 2024-11-10. Relation shown in the video amounts to cl ( 2 t ) = cos 4 2 ( t ) − sin 4 2 ( t ) cos 4 2 ( t ) + sin 4 2 ( t ) {\displaystyle \operatorname {cl} ({\sqrt {2}}t)={\frac {\cos _{4}^{2}(t)-\sin _{4}^{2}(t)}{\cos _{4}^{2}(t)+\sin _{4}^{2}(t)}}} </li></ul> <ul><li><a href="/facts/Niels_Henrik_Abel/6kMfdRkT">Abel, Niels Henrik</a> (1827–1828) "Recherches sur les fonctions elliptiques" [Research on elliptic functions] (in French). <i><a href="/facts/Crelle%27s_Journal/YCcRSaAQ">Crelle's Journal</a></i>.<a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0002?tify={%22pages%22:%5B113%5D}">Part 1</a>. 1827. 2 (2): 101–181. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1827.2.101">10.1515/crll.1827.2.101</a>.<a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0003?tify={%22pages%22:%5B166%5D}">Part 2</a>. 1828. 3 (3): 160–190. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1828.3.160">10.1515/crll.1828.3.160</a>.</li> <li><a href="/facts/Oscar_S._Adams/1ss0vLlI">Adams, Oscar S.</a> (1925). <a href="https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf"><i>Elliptic Functions Applied to Conformal World Maps</i></a> (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112.</li> <li>Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". <i>Archive for History of Exact Sciences</i>. 29 (2): 131–149. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00348244">10.1007/BF00348244</a>.</li> <li>Berndt, Bruce C. (1994). <i>Ramanujan's Notebooks Part IV</i> (First ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-6932-8.</li> <li><a href="/facts/Jonathan_Borwein/j5xCll2M">Borwein, Jonatham M.</a>; <a href="/facts/Peter_Borwein/dJtVFCLg">Borwein, Peter B.</a> (1987). "2.7 The Landen Transformation". <i>Pi and the AGM</i>. Wiley-Interscience. p. 60.</li> <li><a href="/facts/Umberto_Bottazzini/Ek1wao8r">Bottazzini, Umberto</a>; <a href="/facts/Jeremy_Gray/Vbo8J5mk">Gray, Jeremy</a> (2013). <i>Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory</i>. 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Cambridge.</li> <li><a href="/facts/David_A._Cox/y7AQXm0V">Cox, David Archibald</a> (January 1984). <a href="https://www.researchgate.net/publication/248675540">"The Arithmetic-Geometric Mean of Gauss"</a>. <i>L'Enseignement Mathématique</i>. 30 (2): 275–330.</li> <li>Cox, David Archibald; Shurman, Jerry (2005). <a href="https://people.reed.edu/~jerry/Clover/cloverexcerpt.pdf">"Geometry and number theory on clovers"</a> (PDF). <i>The American Mathematical Monthly</i>. 112 (8): 682–704. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F00029890.2005.11920241">10.1080/00029890.2005.11920241</a>.</li> <li>Cox, David Archibald (2012). "The Lemniscate". <i>Galois Theory</i>. Wiley. pp. 463–514. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2F9781118218457.ch15">10.1002/9781118218457.ch15</a>.</li> <li>Cox, David Archibald (2013). <a href="https://dacox.people.amherst.edu/primes.html"><i>Primes of the Form </i><i>x</i>2 + <i>ny</i>2<i></i></a> (Second ed.). 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Nebert. pp. 524–547.</li> <li><a href="/facts/Leonhard_Euler/z2fsbjgj">Euler, Leonhard</a> (1761). <a href="https://archive.org/details/novicommentariia06impe/page/58/">"Observationes de comparatione arcuum curvarum irrectificibilium"</a> [Observations on the comparison of arcs of irrectifiable curves]. <i>Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae</i> (in Latin). 6: 58–84. <a href="/facts/Leonhard_Euler/z2fsbjgj">E</a><a href="https://scholarlycommons.pacific.edu/euler-works/252/">252</a>. (<a href="https://archive.org/details/novicommentariia06impe/page/n617/">Figures</a>)</li> <li>Euler, Leonhard (1786). <a href="https://archive.org/details/actaacademiae82impe/page/34/">"De miris proprietatibus curvae elasticae sub aequatione y = ∫ x x d x / 1 − x 4 {\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}} contentae"</a> [On the amazing properties of elastic curves contained in equation y = ∫ x x d x / 1 − x 4 {\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}} ]. <i>Acta Academiae Scientiarum Imperialis Petropolitanae</i> (in Latin). 1782 (2): 34–61. E <a href="https://scholarlycommons.pacific.edu/euler-works/605/">605</a>.</li> <li>Eymard, Pierre; Lafon, Jean-Pierre (2004). <i>The Number Pi</i>. Translated by Wilson, Stephen. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-3246-8.</li> <li><a href="/facts/Giulio_Carlo_de%27_Toschi_di_Fagnano/9LF6T1oh">Fagnano, Giulio Carlo</a> (1718–1723) "Metodo per misurare la lemniscata" [Method for measuring the lemniscate]. <i>Giornale de' letterati d'Italia</i> (in Italian).<a href="https://archive.org/details/giornaledeletter1718roya/page/258/">"Schediasma primo"</a> [Part 1]. 1718. 29: 258–269.<a href="https://archive.org/details/giornaledeletter1723roya/page/197">"Giunte al primo schediasma"</a> [Addendum to part 1]. 1723. 34: 197–207.<a href="https://archive.org/details/giornaledelette1718roya_0/page/87/">"Schediasma secondo"</a> [Part 2]. 1718. 30: 87–111.Reprinted as Fagnano (1850). <a href="https://archive.org/details/operematfagnano02itws0006/page/n322/">"32–34. Metodo per misurare la lemniscata"</a>. <i>Opere Matematiche, vol. 2</i>. Allerighi e Segati. pp. 293–313. 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MacMillan.</li> <li>Guyou, Émile (1887). <a href="https://www.retronews.fr/journal/annales-hydrographiques/1-janvier-1887/1877/4868382/23">"Nouveau système de projection de la sphère: Généralisation de la projection de Mercator"</a> [New system of projection of the sphere]. <i>Annales Hydrographiques</i>. Série 2 (in French). 9: 16–35.</li> <li>Houzel, Christian (1978). "Fonctions elliptiques et intégrales abéliennes" [Elliptic functions and Abelian integrals]. In <a href="/facts/Jean_Dieudonn%C3%A9/gWagzMdu">Dieudonné, Jean</a> (ed.). <i>Abrégé d'histoire des mathématiques, 1700–1900. II</i> (in French). 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Cambridge. pp. 524–528.</li> <li><a href="/facts/Edmund_T._Whittaker/b7wl3DMJ">Whittaker, Edmund Taylor</a>; <a href="/facts/George_N._Watson/VLfFjgjh">Watson, George Neville</a> (1927) [4th ed. 1927]. "21 The theta functions". <a href="/facts/A_Course_of_Modern_Analysis/RI1vXWBT"><i>A Course of Modern Analysis</i></a> (4th ed.). Cambridge. pp. 469–470.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Fagnano (1718–1723); Euler (1761); Gauss (1917) - Fagnano, Giulio Carlo (1718–1723) "Metodo per misurare la lemniscata" [Method for measuring the lemniscate]. Giornale de' letterati d'Italia (in Italian)."Schediasma primo" [Part 1]. 1718. 29: 258–269."Giunte al primo schediasma" [Addendum to part 1]. 1723. 34: 197–207."Schediasma secondo" [Part 2]. 1718. 30: 87–111.Reprinted as Fagnano (1850). "32–34. Metodo per misurare la lemniscata". Opere Matematiche, vol. 2. Allerighi e Segati. pp. 293–313. (Figures) <a href="https://archive.org/details/giornaledeletter1718roya/page/258/" target="_blank">https://archive.org/details/giornaledeletter1718roya/page/258/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Gauss (1917) p. 199 used the symbols sl and cl for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses sinlem and coslem. Whittaker & Watson (1920) use the symbols sin lemn and cos lemn. Some sources use the generic letters s and c. Prasolov & Solovyev (1997) use the letter φ for the lemniscate sine and φ′ for its derivative. - Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. <a href="https://gdz.sub.uni-goettingen.de/id/PPN236018647" target="_blank">https://gdz.sub.uni-goettingen.de/id/PPN236018647</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>The circle x 2 + y 2 = x {\displaystyle x^{2}+y^{2}=x} is the unit-diameter circle centered at ( 1 2 , 0 ) {\textstyle {\bigl (}{\tfrac {1}{2}},0{\bigr )}} with polar equation r = cos θ , {\displaystyle r=\cos \theta ,} the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} centered at the origin. Notice that the lemniscate ( x 2 + y 2 ) 2 = x 2 − y 2 {\displaystyle {\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}} is the degree-4 clover. - Cox, David Archibald; Shurman, Jerry (2005). "Geometry and number theory on clovers" (PDF). The American Mathematical Monthly. 112 (8): 682–704. doi:10.1080/00029890.2005.11920241. <a href="https://people.reed.edu/~jerry/Clover/cloverexcerpt.pdf" target="_blank">https://people.reed.edu/~jerry/Clover/cloverexcerpt.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>The fundamental periods ( 1 + i ) ϖ {\displaystyle (1+i)\varpi } and ( 1 − i ) ϖ {\displaystyle (1-i)\varpi } are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Robinson (2019a) starts from this definition and thence derives other properties of the lemniscate functions. - Robinson, Paul L. (2019a). "The Lemniscatic Functions". arXiv:1902.08614. <a href="https://arxiv.org/abs/1902.08614" target="_blank">https://arxiv.org/abs/1902.08614</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>This map was the first ever picture of a Schwarz–Christoffel mapping, in Schwarz (1869) p. 113. - Schwarz, Hermann Amandus (1869). "Ueber einige Abbildungsaufgaben" [About some mapping problems]. Crelle's Journal (in German). 70: 105–120. doi:10.1515/crll.1869.70.105. <a href="https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1869_70/page/105" target="_blank">https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1869_70/page/105</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Schappacher (1997). OEIS sequence A062539 lists the lemniscate constant's decimal digits. - Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290. <a href="http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf" target="_blank">http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1997_LemniscProvis.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Levin (2006) - Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. <a href="https://web.archive.org/web/20041220213524id_/http://math.berkeley.edu:80/~adlevin/Lemniscate.pdf" target="_blank">https://web.archive.org/web/20041220213524id_/http://math.berkeley.edu:80/~adlevin/Lemniscate.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Todd (1975) - Todd, John (1975). "The lemniscate constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. <a href="https://doi.org/10.1145%2F360569.360580" target="_blank">https://doi.org/10.1145%2F360569.360580</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Cox (1984) - Cox, David Archibald (January 1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330. <a href="https://www.researchgate.net/publication/248675540" target="_blank">https://www.researchgate.net/publication/248675540</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Combining the first and fourth identity gives sl z = − i / sl ( z − ( 1 + i ) ϖ / 2 ) {\displaystyle \operatorname {sl} z=-i/\operatorname {sl} (z-(1+i)\varpi /2)} . This identity is (incorrectly) given in Eymard & Lafon (2004) p. 226, without the minus sign at the front of the right-hand side. - Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>The even Gaussian integers are the residue class of 0, modulo 1 + i, the black squares on a checkerboard. <a href="/wiki/Checkerboard#Mathematical_description" target="_blank">/wiki/Checkerboard#Mathematical_description</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Prasolov & Solovyev (1997); Robinson (2019a) - Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. <a href="https://doi.org/10.1090%2Fmmono%2F170" target="_blank">https://doi.org/10.1090%2Fmmono%2F170</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Cox (2012) - Cox, David Archibald (2012). "The Lemniscate". Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15. <a href="https://doi.org/10.1002%2F9781118218457.ch15" target="_blank">https://doi.org/10.1002%2F9781118218457.ch15</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Reinhardt & Walker (2010a) §22.12.6, §22.12.12 - Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge. <a href="https://dlmf.nist.gov/22" target="_blank">https://dlmf.nist.gov/22</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Analogously, 1 sin z = ∑ n ∈ Z ( − 1 ) n z + n π . {\displaystyle {\frac {1}{\sin z}}=\sum _{n\in \mathbb {Z} }{\frac {(-1)^{n}}{z+n\pi }}.} <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Lindqvist & Peetre (2001) generalizes the first of these forms. - Lindqvist, Peter; Peetre, Jaak (2001). "Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals" (PDF). The American Mathematical Monthly. 108 (5): 403–410. doi:10.1080/00029890.2001.11919766. <a href="https://web.archive.org/web/20220528124044/https://people.math.osu.edu/lang.162/book/LiPe3.pdf" target="_blank">https://web.archive.org/web/20220528124044/https://people.math.osu.edu/lang.162/book/LiPe3.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Ayoub (1984); Prasolov & Solovyev (1997) - Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244. <a href="https://doi.org/10.1007%2FBF00348244" target="_blank">https://doi.org/10.1007%2FBF00348244</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Euler (1761) §44 p. 79, §47 pp. 80–81 - Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. <a href="https://archive.org/details/novicommentariia06impe/page/58/" target="_blank">https://archive.org/details/novicommentariia06impe/page/58/</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Euler (1761) §46 p. 80 - Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. <a href="https://archive.org/details/novicommentariia06impe/page/58/" target="_blank">https://archive.org/details/novicommentariia06impe/page/58/</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Euler (1761) §46 p. 80 - Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. <a href="https://archive.org/details/novicommentariia06impe/page/58/" target="_blank">https://archive.org/details/novicommentariia06impe/page/58/</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Cox (2012) - Cox, David Archibald (2012). "The Lemniscate". Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15. <a href="https://doi.org/10.1002%2F9781118218457.ch15" target="_blank">https://doi.org/10.1002%2F9781118218457.ch15</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>In fact, i ε = sl β ϖ 2 {\displaystyle i^{\varepsilon }=\operatorname {sl} {\tfrac {\beta \varpi }{2}}} . <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Cox & Hyde (2014) - Cox, David Archibald; Hyde, Trevor (2014). "The Galois theory of the lemniscate" (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006. <a href="http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf" target="_blank">http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Gómez-Molleda & Lario (2019) - Gómez-Molleda, M. A.; Lario, Joan-C. (2019). "Ruler and Compass Constructions of the Equilateral Triangle and Pentagon in the Lemniscate Curve". The Mathematical Intelligencer. 41 (4): 17–21. doi:10.1007/s00283-019-09892-w. <a href="https://doi.org/10.1007%2Fs00283-019-09892-w" target="_blank">https://doi.org/10.1007%2Fs00283-019-09892-w</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>The fourth root with the least positive principal argument is chosen. <a href="/wiki/Argument_(complex_analysis)" target="_blank">/wiki/Argument_(complex_analysis)</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Cox & Hyde (2014) - Cox, David Archibald; Hyde, Trevor (2014). "The Galois theory of the lemniscate" (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006. <a href="http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf" target="_blank">http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>The restriction to positive and odd β {\displaystyle \beta } can be dropped in deg Λ β = | ( O / β O ) × | {\displaystyle \operatorname {deg} \Lambda _{\beta }=\left|({\mathcal {O}}/\beta {\mathcal {O}})^{\times }\right|} . <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Cox (2013) p. 142, Example 7.29(c) - Cox, David Archibald (2013). Primes of the Form x2 + ny2 (Second ed.). Wiley. <a href="https://dacox.people.amherst.edu/primes.html" target="_blank">https://dacox.people.amherst.edu/primes.html</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Rosen (1981) - Rosen, Michael (1981). "Abel's Theorem on the Lemniscate". The American Mathematical Monthly. 88 (6): 387–395. doi:10.2307/2321821. <a href="https://doi.org/10.2307%2F2321821" target="_blank">https://doi.org/10.2307%2F2321821</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Eymard & Lafon (2004) p. 200 - Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>And the area enclosed by L {\displaystyle {\mathcal {L}}} is 1 {\displaystyle 1} , which stands in stark contrast to the unit circle (whose enclosed area is a non-constructible number). <a href="/wiki/Squaring_the_circle" target="_blank">/wiki/Squaring_the_circle</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Euler (1761); Siegel (1969). Prasolov & Solovyev (1997) use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same. - Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. <a href="https://archive.org/details/novicommentariia06impe/page/58/" target="_blank">https://archive.org/details/novicommentariia06impe/page/58/</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Reinhardt & Walker (2010a) §22.18.E6 - Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge. <a href="https://dlmf.nist.gov/22" target="_blank">https://dlmf.nist.gov/22</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Siegel (1969); Schappacher (1997) - Siegel, Carl Ludwig (1969). "1. Elliptic Functions". Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0. <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Such numbers are OEIS sequence A003401. <a href="//oeis.org/A003401" target="_blank">//oeis.org/A003401</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Abel (1827–1828); Rosen (1981); Prasolov & Solovyev (1997) - Abel, Niels Henrik (1827–1828) "Recherches sur les fonctions elliptiques" [Research on elliptic functions] (in French). Crelle's Journal.Part 1. 1827. 2 (2): 101–181. doi:10.1515/crll.1827.2.101.Part 2. 1828. 3 (3): 160–190. doi:10.1515/crll.1828.3.160. <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0002?tify={%22pages%22:%5B113%5D}" target="_blank">https://gdz.sub.uni-goettingen.de/id/PPN243919689_0002?tify={%22pages%22:%5B113%5D}</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Euler (1786); Sridharan (2004); Levien (2008) - Euler, Leonhard (1786). "De miris proprietatibus curvae elasticae sub aequatione y = ∫ x x d x / 1 − x 4 {\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}} contentae" [On the amazing properties of elastic curves contained in equation y = ∫ x x d x / 1 − x 4 {\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}} ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. <a href="https://archive.org/details/actaacademiae82impe/page/34/" target="_blank">https://archive.org/details/actaacademiae82impe/page/34/</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>"A104203". The On-Line Encyclopedia of Integer Sequences. <a href="https://oeis.org/A104203" target="_blank">https://oeis.org/A104203</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. pp. 12, 44. ISBN 1-58488-210-7. <a href="1-58488-210-7" target="_blank">1-58488-210-7</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>"A193543 - Oeis". <a href="https://oeis.org/A193543" target="_blank">https://oeis.org/A193543</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. ISBN 1-58488-210-7. p. 79, eq. 5.36 <a href="1-58488-210-7" target="_blank">1-58488-210-7</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Lomont, J.S.; Brillhart, John (2001). Elliptic Polynomials. CRC Press. ISBN 1-58488-210-7. p. 79, eq. 5. 36 and p. 78, eq. 5.33 <a href="1-58488-210-7" target="_blank">1-58488-210-7</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></p></li> <li id="fn:44"><p>"A193543 - Oeis". <a href="https://oeis.org/A193543" target="_blank">https://oeis.org/A193543</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></p></li> <li id="fn:45"><p>"A193543 - Oeis". <a href="https://oeis.org/A193543" target="_blank">https://oeis.org/A193543</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></p></li> <li id="fn:46"><p>"A289695 - Oeis". <a href="https://oeis.org/A289695" target="_blank">https://oeis.org/A289695</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></p></li> <li id="fn:47"><p>Wall, H. S. (1948). Analytic Theory of Continued Fractions. Chelsea Publishing Company. pp. 374–375. <a href="#fnref:47" class="footnote-back-ref">↩</a></p></li> <li id="fn:48"><p>Reinhardt & Walker (2010a) §22.12.12 - Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge. <a href="https://dlmf.nist.gov/22" target="_blank">https://dlmf.nist.gov/22</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></p></li> <li id="fn:49"><p>In general, sinh ( x − n π ) {\displaystyle \sinh(x-n\pi )} and sin ( x − n π i ) = − i sinh ( i x + n π ) {\displaystyle \sin(x-n\pi i)=-i\sinh(ix+n\pi )} are not equivalent, but the resulting infinite sum is the same. <a href="#fnref:49" class="footnote-back-ref">↩</a></p></li> <li id="fn:50"><p>Reinhardt & Walker (2010a) §22.11 - Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge. <a href="https://dlmf.nist.gov/22" target="_blank">https://dlmf.nist.gov/22</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></p></li> <li id="fn:51"><p>Reinhardt & Walker (2010a) §22.2.E7 - Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge. <a href="https://dlmf.nist.gov/22" target="_blank">https://dlmf.nist.gov/22</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></p></li> <li id="fn:52"><p>Berndt (1994) p. 247, 248, 253 - Berndt, Bruce C. (1994). Ramanujan's Notebooks Part IV (First ed.). Springer. ISBN 978-1-4612-6932-8. <a href="#fnref:52" class="footnote-back-ref">↩</a></p></li> <li id="fn:53"><p>"A193543 - Oeis". <a href="https://oeis.org/A193543" target="_blank">https://oeis.org/A193543</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></p></li> <li id="fn:54"><p>"A289695 - Oeis". <a href="https://oeis.org/A289695" target="_blank">https://oeis.org/A289695</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></p></li> <li id="fn:55"><p>Reinhardt & Walker (2010a) §22.11.E1 - Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge. <a href="https://dlmf.nist.gov/22" target="_blank">https://dlmf.nist.gov/22</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></p></li> <li id="fn:56"><p>Whittaker & Watson (1927) - Whittaker, Edmund Taylor; Watson, George Neville (1927) [4th ed. 1927]. "21 The theta functions". A Course of Modern Analysis (4th ed.). Cambridge. pp. 469–470. <a href="#fnref:56" class="footnote-back-ref">↩</a></p></li> <li id="fn:57"><p>Borwein & Borwein (1987) - Borwein, Jonatham M.; Borwein, Peter B. (1987). "2.7 The Landen Transformation". Pi and the AGM. Wiley-Interscience. p. 60. <a href="#fnref:57" class="footnote-back-ref">↩</a></p></li> <li id="fn:58"><p>Eymard & Lafon (2004) p. 227. - Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. <a href="#fnref:58" class="footnote-back-ref">↩</a></p></li> <li id="fn:59"><p>Eymard & Lafon (2004) p. 227. - Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. <a href="#fnref:59" class="footnote-back-ref">↩</a></p></li> <li id="fn:60"><p>Cartan, H. (1961). Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes (in French). Hermann. pp. 160–164. <a href="#fnref:60" class="footnote-back-ref">↩</a></p></li> <li id="fn:61"><p>More precisely, suppose { a n } {\displaystyle \{a_{n}\}} is a sequence of bounded complex functions on a set S {\displaystyle S} , such that ∑ | a n ( z ) | {\textstyle \sum \left|a_{n}(z)\right|} converges uniformly on S {\displaystyle S} . If { n 1 , n 2 , n 3 , … } {\displaystyle \{n_{1},n_{2},n_{3},\ldots \}} is any permutation of { 1 , 2 , 3 , … } {\displaystyle \{1,2,3,\ldots \}} , then ∏ n = 1 ∞ ( 1 + a n ( z ) ) = ∏ k = 1 ∞ ( 1 + a n k ( z ) ) {\textstyle \prod _{n=1}^{\infty }(1+a_{n}(z))=\prod _{k=1}^{\infty }(1+a_{n_{k}}(z))} for all z ∈ S {\displaystyle z\in S} . The theorem in question then follows from the fact that there exists a bijection between the natural numbers and α {\displaystyle \alpha } 's (resp. β {\displaystyle \beta } 's). <a href="/wiki/Permutation" target="_blank">/wiki/Permutation</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></p></li> <li id="fn:62"><p>Bottazzini & Gray (2013) p. 58 - Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1. <a href="https://doi.org/10.1007%2F978-1-4614-5725-1" target="_blank">https://doi.org/10.1007%2F978-1-4614-5725-1</a> <a href="#fnref:62" class="footnote-back-ref">↩</a></p></li> <li id="fn:63"><p>More precisely, if for each k {\displaystyle k} , lim n → ∞ a k ( n ) {\textstyle \lim _{n\to \infty }a_{k}(n)} exists and there is a convergent series ∑ k = 1 ∞ M k {\textstyle \sum _{k=1}^{\infty }M_{k}} of nonnegative real numbers such that | a k ( n ) | ≤ M k {\displaystyle \left|a_{k}(n)\right|\leq M_{k}} for all n ∈ N {\displaystyle n\in \mathbb {N} } and 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} , then lim n → ∞ ∑ k = 1 n a k ( n ) = ∑ k = 1 ∞ lim n → ∞ a k ( n ) . {\displaystyle \lim _{n\to \infty }\sum _{k=1}^{n}a_{k}(n)=\sum _{k=1}^{\infty }\lim _{n\to \infty }a_{k}(n).} <a href="#fnref:63" class="footnote-back-ref">↩</a></p></li> <li id="fn:64"><p>Alternatively, it can be inferred that these expansions exist just from the analyticity of M {\displaystyle M} and N {\displaystyle N} . However, establishing the connection to "multiplying out and collecting like powers" reveals identities between sums of reciprocals and the coefficients of the power series, like ∑ α 1 α 4 = − the coefficient of z 5 {\textstyle \sum _{\alpha }{\frac {1}{\alpha ^{4}}}=-\,{\text{the coefficient of}}\,z^{5}} in the M {\displaystyle M} series, and infinitely many others. <a href="#fnref:64" class="footnote-back-ref">↩</a></p></li> <li id="fn:65"><p>Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 405; there's an error on the page: the coefficient of φ 17 {\displaystyle \varphi ^{17}} should be 107 7 410 154 752 000 {\displaystyle {\tfrac {107}{7\,410\,154\,752\,000}}} , not 107 207 484 333 056 000 {\displaystyle {\tfrac {107}{207\,484\,333\,056\,000}}} . <a href="https://gdz.sub.uni-goettingen.de/id/PPN235999628" target="_blank">https://gdz.sub.uni-goettingen.de/id/PPN235999628</a> <a href="#fnref:65" class="footnote-back-ref">↩</a></p></li> <li id="fn:66"><p>If M ( z ) = ∑ n = 0 ∞ a n z n + 1 {\textstyle M(z)=\sum _{n=0}^{\infty }a_{n}z^{n+1}} , then the coefficients a n {\displaystyle a_{n}} are given by the recurrence a n + 1 = − 1 n + 1 ∑ k = 0 n 2 n − k + 1 a k H n − k + 1 ( n − k + 1 ) ! {\textstyle a_{n+1}=-{\frac {1}{n+1}}\sum _{k=0}^{n}2^{n-k+1}a_{k}{\frac {\mathrm {H} _{n-k+1}}{(n-k+1)!}}} with a 0 = 1 {\displaystyle a_{0}=1} where H n {\displaystyle \mathrm {H} _{n}} are the Hurwitz numbers defined in Lemniscate elliptic functions § Hurwitz numbers. <a href="#fnref:66" class="footnote-back-ref">↩</a></p></li> <li id="fn:67"><p>The power series expansions of M {\displaystyle M} and N {\displaystyle N} are useful for finding a β {\displaystyle \beta } -division polynomial for the β {\displaystyle \beta } -division of the lemniscate L {\displaystyle {\mathcal {L}}} (where β = m + n i {\displaystyle \beta =m+ni} where m , n ∈ Z {\displaystyle m,n\in \mathbb {Z} } such that m + n {\displaystyle m+n} is odd). For example, suppose we want to find a 3 {\displaystyle 3} -division polynomial. Given that M ( 3 z ) = d 9 M ( z ) 9 + d 5 M ( z ) 5 N ( z ) 4 + d 1 M ( z ) N ( z ) 8 {\displaystyle M(3z)=d_{9}M(z)^{9}+d_{5}M(z)^{5}N(z)^{4}+d_{1}M(z)N(z)^{8}} for some constants d 1 , d 5 , d 9 {\displaystyle d_{1},d_{5},d_{9}} , from 3 z − 2 ( 3 z ) 5 5 ! − 36 ( 3 z ) 9 9 ! + O ( z 13 ) = d 9 x 9 + d 5 x 5 y 4 + d 1 x y 8 , {\displaystyle 3z-2{\frac {(3z)^{5}}{5!}}-36{\frac {(3z)^{9}}{9!}}+\operatorname {O} (z^{13})=d_{9}x^{9}+d_{5}x^{5}y^{4}+d_{1}xy^{8},} where x = z − 2 z 5 5 ! − 36 z 9 9 ! + O ( z 13 ) , y = 1 + 2 z 4 4 ! − 4 z 8 8 ! + O ( z 12 ) , {\displaystyle x=z-2{\frac {z^{5}}{5!}}-36{\frac {z^{9}}{9!}}+\operatorname {O} (z^{13}),\quad y=1+2{\frac {z^{4}}{4!}}-4{\frac {z^{8}}{8!}}+\operatorname {O} (z^{12}),} we have { d 1 , d 5 , d 9 } = { 3 , − 6 , − 1 } . {\displaystyle \{d_{1},d_{5},d_{9}\}=\{3,-6,-1\}.} Therefore, a 3 {\displaystyle 3} -division polynomial is − X 9 − 6 X 5 + 3 X {\displaystyle -X^{9}-6X^{5}+3X} (meaning one of its roots is sl ( 2 ϖ / 3 ) {\displaystyle \operatorname {sl} (2\varpi /3)} ). The equations arrived at by this process are the lemniscate analogs of X n = 1 {\displaystyle X^{n}=1} (so that e 2 π i / n {\displaystyle e^{2\pi i/n}} is one of the solutions) which comes up when dividing the unit circle into n {\displaystyle n} arcs of equal length. In the following note, the first few coefficients of the monic normalization of such β {\displaystyle \beta } -division polynomials are described symbolically in terms of β {\displaystyle \beta } . <a href="#fnref:67" class="footnote-back-ref">↩</a></p></li> <li id="fn:68"><p>By utilizing the power series expansion of the N {\displaystyle N} function, it can be proved that a polynomial having sl ( 2 ϖ / β ) {\displaystyle \operatorname {sl} (2\varpi /\beta )} as one of its roots (with β {\displaystyle \beta } from the previous note) is ∑ n = 0 ( β β ¯ − 1 ) / 4 a 4 n + 1 ( β ) X β β ¯ − 4 n {\displaystyle \sum _{n=0}^{(\beta {\overline {\beta }}-1)/4}a_{4n+1}(\beta )X^{\beta {\overline {\beta }}-4n}} where a 1 ( β ) = 1 , a 5 ( β ) = β 4 − β β ¯ 12 , a 9 ( β ) = − β 8 − 70 β 5 β ¯ + 336 β 4 + 35 β 2 β ¯ 2 − 300 β β ¯ 10080 {\displaystyle {\begin{aligned}a_{1}(\beta )&=1,\\a_{5}(\beta )&={\frac {\beta ^{4}-\beta {\overline {\beta }}}{12}},\\a_{9}(\beta )&={\frac {-\beta ^{8}-70\beta ^{5}{\overline {\beta }}+336\beta ^{4}+35\beta ^{2}{\overline {\beta }}^{2}-300\beta {\overline {\beta }}}{10080}}\end{aligned}}} and so on. <a href="#fnref:68" class="footnote-back-ref">↩</a></p></li> <li id="fn:69"><p>Zhuravskiy, A. M. (1941). Spravochnik po ellipticheskim funktsiyam (in Russian). Izd. Akad. Nauk. U.S.S.R. <a href="#fnref:69" class="footnote-back-ref">↩</a></p></li> <li id="fn:70"><p>For example, by the quasi-addition formulas, the duplication formulas and the Pythagorean-like identities, we have M ( 3 z ) = − M ( z ) 9 − 6 M ( z ) 5 N ( z ) 4 + 3 M ( z ) N ( z ) 8 , {\displaystyle M(3z)=-M(z)^{9}-6M(z)^{5}N(z)^{4}+3M(z)N(z)^{8},} N ( 3 z ) = N ( z ) 9 + 6 M ( z ) 4 N ( z ) 5 − 3 M ( z ) 8 N ( z ) , {\displaystyle N(3z)=N(z)^{9}+6M(z)^{4}N(z)^{5}-3M(z)^{8}N(z),} so sl 3 z = − M ( z ) 9 − 6 M ( z ) 5 N ( z ) 4 + 3 M ( z ) N ( z ) 8 N ( z ) 9 + 6 M ( z ) 4 N ( z ) 5 − 3 M ( z ) 8 N ( z ) . {\displaystyle \operatorname {sl} 3z={\frac {-M(z)^{9}-6M(z)^{5}N(z)^{4}+3M(z)N(z)^{8}}{N(z)^{9}+6M(z)^{4}N(z)^{5}-3M(z)^{8}N(z)}}.} On dividing the numerator and the denominator by N ( z ) 9 {\displaystyle N(z)^{9}} , we obtain the triplication formula for sl {\displaystyle \operatorname {sl} } : sl 3 z = − sl 9 z − 6 sl 5 z + 3 sl z 1 + 6 sl 4 z − 3 sl 8 z . {\displaystyle \operatorname {sl} 3z={\frac {-\operatorname {sl} ^{9}z-6\operatorname {sl} ^{5}z+3\operatorname {sl} z}{1+6\operatorname {sl} ^{4}z-3\operatorname {sl} ^{8}z}}.} <a href="#fnref:70" class="footnote-back-ref">↩</a></p></li> <li id="fn:71"><p>Gauss (1866), p. 408 <a href="#fnref:71" class="footnote-back-ref">↩</a></p></li> <li id="fn:72"><p>Robinson (2019a) - Robinson, Paul L. (2019a). "The Lemniscatic Functions". arXiv:1902.08614. <a href="https://arxiv.org/abs/1902.08614" target="_blank">https://arxiv.org/abs/1902.08614</a> <a href="#fnref:72" class="footnote-back-ref">↩</a></p></li> <li id="fn:73"><p>Eymard & Lafon (2004) p. 234 - Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. <a href="#fnref:73" class="footnote-back-ref">↩</a></p></li> <li id="fn:74"><p>Armitage, J. V.; Eberlein, W. F. (2006). Elliptic Functions. Cambridge University Press. p. 49. ISBN 978-0-521-78563-1. <a href="978-0-521-78563-1" target="_blank">978-0-521-78563-1</a> <a href="#fnref:74" class="footnote-back-ref">↩</a></p></li> <li id="fn:75"><p>The identity cl z = cn ( 2 z ; 1 2 ) {\displaystyle \operatorname {cl} z={\operatorname {cn} }\left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right)} can be found in Greenhill (1892) p. 33. - Greenhill, Alfred George (1892). The Applications of Elliptic Functions. MacMillan. <a href="https://archive.org/details/applicationselli00greerich/page/n5/" target="_blank">https://archive.org/details/applicationselli00greerich/page/n5/</a> <a href="#fnref:75" class="footnote-back-ref">↩</a></p></li> <li id="fn:76"><p>Siegel (1969) - Siegel, Carl Ludwig (1969). "1. Elliptic Functions". Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0. <a href="#fnref:76" class="footnote-back-ref">↩</a></p></li> <li id="fn:77"><p>Sloane, N. J. A. (ed.). "Sequence A175576". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:77" class="footnote-back-ref">↩</a></p></li> <li id="fn:78"><p>Berndt, Bruce C. (1989). Ramanujan's Notebooks Part II. Springer. ISBN 978-1-4612-4530-8. p. 96 <a href="978-1-4612-4530-8" target="_blank">978-1-4612-4530-8</a> <a href="#fnref:78" class="footnote-back-ref">↩</a></p></li> <li id="fn:79"><p>Levin (2006); Robinson (2019b) - Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. <a href="https://web.archive.org/web/20041220213524id_/http://math.berkeley.edu:80/~adlevin/Lemniscate.pdf" target="_blank">https://web.archive.org/web/20041220213524id_/http://math.berkeley.edu:80/~adlevin/Lemniscate.pdf</a> <a href="#fnref:79" class="footnote-back-ref">↩</a></p></li> <li id="fn:80"><p>Cox & Hyde (2014) - Cox, David Archibald; Hyde, Trevor (2014). "The Galois theory of the lemniscate" (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006. <a href="http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf" target="_blank">http://www-personal.umich.edu/~tghyde/Cox,%20Hyde%20--%20Galois%20theory%20on%20the%20lemniscate.pdf</a> <a href="#fnref:80" class="footnote-back-ref">↩</a></p></li> <li id="fn:81"><p>Cox (2012) p. 508, 509 - Cox, David Archibald (2012). "The Lemniscate". Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15. <a href="https://doi.org/10.1002%2F9781118218457.ch15" target="_blank">https://doi.org/10.1002%2F9781118218457.ch15</a> <a href="#fnref:81" class="footnote-back-ref">↩</a></p></li> <li id="fn:82"><p>Cox (2012) p. 508, 509 - Cox, David Archibald (2012). "The Lemniscate". Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15. <a href="https://doi.org/10.1002%2F9781118218457.ch15" target="_blank">https://doi.org/10.1002%2F9781118218457.ch15</a> <a href="#fnref:82" class="footnote-back-ref">↩</a></p></li> <li id="fn:83"><p>Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203—206 <a href="978-4-431-54918-5" target="_blank">978-4-431-54918-5</a> <a href="#fnref:83" class="footnote-back-ref">↩</a></p></li> <li id="fn:84"><p>Equivalently, H n = − lim z → 0 d n d z n ( ( 1 + i ) z / 2 sl ( ( 1 + i ) z / 2 ) + z 2 E ( z 2 ; i ) ) {\displaystyle \mathrm {H} _{n}=-\lim _{z\to 0}{\frac {\mathrm {d} ^{n}}{\mathrm {d} z^{n}}}\left({\frac {(1+i)z/2}{\operatorname {sl} ((1+i)z/2)}}+{\frac {z}{2}}{\mathcal {E}}\left({\frac {z}{2}};i\right)\right)} where n ≥ 4 {\displaystyle n\geq 4} and E ( ⋅ ; i ) {\displaystyle {\mathcal {E}}(\cdot ;i)} is the Jacobi epsilon function with modulus i {\displaystyle i} . <a href="/wiki/Jacobi_elliptic_functions#Definition_in_terms_of_inverses_of_elliptic_integrals" target="_blank">/wiki/Jacobi_elliptic_functions#Definition_in_terms_of_inverses_of_elliptic_integrals</a> <a href="#fnref:84" class="footnote-back-ref">↩</a></p></li> <li id="fn:85"><p>The Bernoulli numbers can be determined by an analogous recurrence: B 2 n = − 1 2 n + 1 ∑ k = 1 n − 1 ( 2 n 2 k ) B 2 k B 2 ( n − k ) {\displaystyle \mathrm {B} _{2n}=-{\frac {1}{2n+1}}\sum _{k=1}^{n-1}{\binom {2n}{2k}}\mathrm {B} _{2k}\mathrm {B} _{2(n-k)}} where n ≥ 2 {\displaystyle n\geq 2} and B 2 = 1 / 6 {\displaystyle \mathrm {B} _{2}=1/6} . <a href="#fnref:85" class="footnote-back-ref">↩</a></p></li> <li id="fn:86"><p>Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203—206 <a href="978-4-431-54918-5" target="_blank">978-4-431-54918-5</a> <a href="#fnref:86" class="footnote-back-ref">↩</a></p></li> <li id="fn:87"><p>Katz, Nicholas M. (1975). "The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers". Mathematische Annalen. 216 (1): 1–4. See eq. (9) <a href="https://link.springer.com/article/10.1007/BF02547966" target="_blank">https://link.springer.com/article/10.1007/BF02547966</a> <a href="#fnref:87" class="footnote-back-ref">↩</a></p></li> <li id="fn:88"><p>For more on the ν {\displaystyle \nu } function, see Lemniscate constant. <a href="/wiki/Lemniscate_constant" target="_blank">/wiki/Lemniscate_constant</a> <a href="#fnref:88" class="footnote-back-ref">↩</a></p></li> <li id="fn:89"><p>Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203—206 <a href="978-4-431-54918-5" target="_blank">978-4-431-54918-5</a> <a href="#fnref:89" class="footnote-back-ref">↩</a></p></li> <li id="fn:90"><p>Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203—206 <a href="978-4-431-54918-5" target="_blank">978-4-431-54918-5</a> <a href="#fnref:90" class="footnote-back-ref">↩</a></p></li> <li id="fn:91"><p>Hurwitz, Adolf (1963). Mathematische Werke: Band II (in German). Springer Basel AG. p. 370 <a href="#fnref:91" class="footnote-back-ref">↩</a></p></li> <li id="fn:92"><p>Arakawa et al. (2014) define H 4 n {\displaystyle \mathrm {H} _{4n}} by the expansion of 1 / sl 2 . {\displaystyle 1/\operatorname {sl} ^{2}.} <a href="#fnref:92" class="footnote-back-ref">↩</a></p></li> <li id="fn:93"><p>Eisenstein, G. (1846). "Beiträge zur Theorie der elliptischen Functionen". Journal für die reine und angewandte Mathematik (in German). 30. Eisenstein uses φ = sl {\displaystyle \varphi =\operatorname {sl} } and ω = 2 ϖ {\displaystyle \omega =2\varpi } . <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0030?tify=%7B%22pages%22%3A%5B202%5D%2C%22view%22%3A%22scan%22%7D" target="_blank">https://gdz.sub.uni-goettingen.de/id/PPN243919689_0030?tify=%7B%22pages%22%3A%5B202%5D%2C%22view%22%3A%22scan%22%7D</a> <a href="#fnref:93" class="footnote-back-ref">↩</a></p></li> <li id="fn:94"><p>Ogawa, Takuma (2005). "Similarities between the trigonometric function and the lemniscate function from arithmetic view point". Tsukuba Journal of Mathematics. 29 (1). <a href="https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full" target="_blank">https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full</a> <a href="#fnref:94" class="footnote-back-ref">↩</a></p></li> <li id="fn:95"><p>Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. - Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. <a href="https://archive.org/details/sim_american-journal-of-mathematics_1879_2/page/n403/mode/2up" target="_blank">https://archive.org/details/sim_american-journal-of-mathematics_1879_2/page/n403/mode/2up</a> <a href="#fnref:95" class="footnote-back-ref">↩</a></p></li> <li id="fn:96"><p>Adams (1925) - Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. <a href="https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf" target="_blank">https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf</a> <a href="#fnref:96" class="footnote-back-ref">↩</a></p></li> <li id="fn:97"><p>Adams (1925); Lee (1976). - Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. <a href="https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf" target="_blank">https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_112.pdf</a> <a href="#fnref:97" class="footnote-back-ref">↩</a></p></li> <li id="fn:98"><p>Rančić, Purser & Mesinger (1996); McGregor (2005). - Rančić, Miodrag; Purser, R. James; Mesinger, Fedor (1996). "A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates". Quarterly Journal of the Royal Meteorological Society. 122 (532): 959–982. doi:10.1002/qj.49712253209. <a href="https://doi.org/10.1002%2Fqj.49712253209" target="_blank">https://doi.org/10.1002%2Fqj.49712253209</a> <a href="#fnref:98" class="footnote-back-ref">↩</a></p></li> </ol>