Inverse hyperbolic functions

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the inverse hyperbolic functions are <a href="/facts/Inverse_function/Tg5nnXlu">inverses</a> of the <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic functions</a>, analogous to the <a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">inverse circular functions</a>. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar- or with a superscript 
 
 
 
 
 −
 1
 
 
 
 {\displaystyle {-1}}
 
 (for example arcsinh, arsinh, or 
 
 
 
 
 sinh
 
 −
 1
 
 
 
 
 {\displaystyle \sinh ^{-1}}
 
).
For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding <a href="/facts/Hyperbolic_angle/m4IaIh0g">hyperbolic angle measure</a>, for example 
 
 
 
 arsinh
 ⁡
 (
 sinh
 ⁡
 a
 )
 =
 a
 
 
 {\displaystyle \operatorname {arsinh} (\sinh a)=a}
 
 and 
 
 
 
 sinh
 ⁡
 (
 arsinh
 ⁡
 x
 )
 =
 x
 .
 
 
 {\displaystyle \sinh(\operatorname {arsinh} x)=x.}
 
 Hyperbolic angle measure is the <a href="/facts/Arc_length/QgTrmLxY">length of an arc</a> of a <a href="/facts/Unit_hyperbola/AvwG6gJ9">unit hyperbola</a> 
 
 
 
 
 x
 
 2
 
 
 −
 
 y
 
 2
 
 
 =
 1
 
 
 {\displaystyle x^{2}-y^{2}=1}
 
 as measured in the Lorentzian plane (not the length of a hyperbolic arc in the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a>), and twice the <a href="/facts/Area/KsNG1G9t">area</a> of the corresponding <a href="/facts/Hyperbolic_sector/PCEFy8fd">hyperbolic sector</a>. This is analogous to the way <a href="/facts/Angle/gFUTEQYq">circular angle measure</a> is the arc length of an arc of the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> in the Euclidean plane or twice the area of the corresponding <a href="/facts/Circular_sector/TqwCaEA5">circular sector</a>. Alternately hyperbolic angle is the area of a sector of the hyperbola 
 
 
 
 x
 y
 =
 1.
 
 
 {\displaystyle xy=1.}
 
 Some authors call the inverse hyperbolic functions hyperbolic area functions.
Hyperbolic functions occur in the calculation of angles and distances in <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>. They also occur in the solutions of many linear <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> (such as the equation defining a <a href="/facts/Catenary/fWVU770E">catenary</a>), <a href="/facts/Cubic_function/t67bqD6c">cubic equations</a>, and <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a> in <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a>. <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equations</a> are important in many areas of <a href="/facts/Physics/a7aH2y08">physics</a>, including <a href="/facts/Electromagnetic_theory/oAkmALHT">electromagnetic theory</a>, <a href="/facts/Heat_transfer/1IV3WbFL">heat transfer</a>, <a href="/facts/Fluid_dynamics/y7uzSCfg">fluid dynamics</a>, and <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>.

Inverse hyperbolic functions open-in-new

Inverse hyperbolic functions