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Kronecker delta
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<h2 id="properties">Properties</h2> <p>The following equations are satisfied: ∑ j δ i j a j = a i , ∑ i a i δ i j = a j , ∑ k δ i k δ k j = δ i j . {\displaystyle {\begin{aligned}\sum _{j}\delta _{ij}a_{j}&=a_{i},\\\sum _{i}a_{i}\delta _{ij}&=a_{j},\\\sum _{k}\delta _{ik}\delta _{kj}&=\delta _{ij}.\end{aligned}}} Therefore, the matrix δ can be considered as an identity matrix. </p><p>Another useful representation is the following form: δ n m = lim N → ∞ 1 N ∑ k = 1 N e 2 π i k N ( n − m ) {\displaystyle \delta _{nm}=\lim _{N\to \infty }{\frac {1}{N}}\sum _{k=1}^{N}e^{2\pi i{\frac {k}{N}}(n-m)}} This can be derived using the formula for the <a href="/facts/Geometric_series/Ne5TBH57">geometric series</a>. </p> <h2 id="alternative-notation">Alternative notation</h2> <p>Using the <a href="/facts/Iverson_bracket/HUuSwKe8">Iverson bracket</a>: δ i j = [ i = j ] . {\displaystyle \delta _{ij}=[i=j].} </p><p>Often, a single-argument notation δ i {\displaystyle \delta _{i}} is used, which is equivalent to setting j = 0 {\displaystyle j=0} : δ i = δ i 0 = { 0 , if i ≠ 0 1 , if i = 0 {\displaystyle \delta _{i}=\delta _{i0}={\begin{cases}0,&{\text{if }}i\neq 0\\1,&{\text{if }}i=0\end{cases}}} </p><p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, it can be thought of as a <a href="/facts/Tensor/pNjFo5tV">tensor</a>, and is written δ j i {\displaystyle \delta _{j}^{i}} . Sometimes the Kronecker delta is called the substitution tensor.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="digital-signal-processing">Digital signal processing</h2> <p>In the study of <a href="/facts/Digital_signal_processing/PEG7azr8">digital signal processing</a> (DSP), the unit sample function δ [ n ] {\displaystyle \delta [n]} represents a special case of a 2-dimensional Kronecker delta function δ i j {\displaystyle \delta _{ij}} where the Kronecker indices include the number zero, and where one of the indices is zero. In this case: δ [ n ] ≡ δ n 0 ≡ δ 0 n where − ∞ < n < ∞ {\displaystyle \delta [n]\equiv \delta _{n0}\equiv \delta _{0n}~~~{\text{where}}-\infty <n<\infty } </p><p>Or more generally where: δ [ n − k ] ≡ δ [ k − n ] ≡ δ n k ≡ δ k n where − ∞ < n < ∞ , − ∞ < k < ∞ {\displaystyle \delta [n-k]\equiv \delta [k-n]\equiv \delta _{nk}\equiv \delta _{kn}{\text{where}}-\infty <n<\infty ,-\infty <k<\infty } </p><p>However, this is only a special case. In tensor calculus, it is more common to number basis vectors in a particular dimension starting with index 1, rather than index 0. In this case, the relation δ [ n ] ≡ δ n 0 ≡ δ 0 n {\displaystyle \delta [n]\equiv \delta _{n0}\equiv \delta _{0n}} does not exist, and in fact, the Kronecker delta function and the unit sample function are different functions that overlap in the specific case where the indices include the number 0, the number of indices is 2, and one of the indices has the value of zero. </p><p>While the discrete unit sample function and the Kronecker delta function use the same letter, they differ in the following ways. For the discrete unit sample function, it is more conventional to place a single integer index in square braces; in contrast the Kronecker delta can have any number of indexes. Further, the purpose of the discrete unit sample function is different from the Kronecker delta function. In DSP, the discrete unit sample function is typically used as an input function to a discrete system for discovering the system function of the system which will be produced as an output of the system. In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an <a href="/facts/Einstein_summation_convention/UXwS43wU">Einstein summation convention</a>. </p><p>The discrete unit sample function is more simply defined as: δ [ n ] = { 1 n = 0 0 n is another integer {\displaystyle \delta [n]={\begin{cases}1&n=0\\0&n{\text{ is another integer}}\end{cases}}} </p><p>In addition, the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as: { ∫ − ε + ε δ ( t ) d t = 1 ∀ ε > 0 δ ( t ) = 0 ∀ t ≠ 0 {\displaystyle {\begin{cases}\int _{-\varepsilon }^{+\varepsilon }\delta (t)dt=1&\forall \varepsilon >0\\\delta (t)=0&\forall t\neq 0\end{cases}}} </p><p>Unlike the Kronecker delta function δ i j {\displaystyle \delta _{ij}} and the unit sample function δ [ n ] {\displaystyle \delta [n]} , the Dirac delta function δ ( t ) {\displaystyle \delta (t)} does not have an integer index, it has a single continuous non-integer value t. </p><p>To confuse matters more, the <a href="/facts/Unit_impulse_function/V8PjRP8T">unit impulse function</a> is sometimes used to refer to either the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> δ ( t ) {\displaystyle \delta (t)} , or the unit sample function δ [ n ] {\displaystyle \delta [n]} . </p> <h2 id="notable-properties">Notable properties</h2> <p>The Kronecker delta has the so-called <i>sifting</i> property that for j ∈ Z {\displaystyle j\in \mathbb {Z} } : ∑ i = − ∞ ∞ a i δ i j = a j . {\displaystyle \sum _{i=-\infty }^{\infty }a_{i}\delta _{ij}=a_{j}.} and if the integers are viewed as a <a href="/facts/Measure_space/w9Dx4KY1">measure space</a>, endowed with the <a href="/facts/Counting_measure/TA4u39yF">counting measure</a>, then this property coincides with the defining property of the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> ∫ − ∞ ∞ δ ( x − y ) f ( x ) d x = f ( y ) , {\displaystyle \int _{-\infty }^{\infty }\delta (x-y)f(x)\,dx=f(y),} and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, δ ( t ) {\displaystyle \delta (t)} generally indicates continuous time (Dirac), whereas arguments like i {\displaystyle i} , j {\displaystyle j} , k {\displaystyle k} , l {\displaystyle l} , m {\displaystyle m} , and n {\displaystyle n} are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: δ [ n ] {\displaystyle \delta [n]} . The Kronecker delta is not the result of directly sampling the Dirac delta function. </p><p>The Kronecker delta forms the multiplicative <a href="/facts/Identity_element/9nss3xHY">identity element</a> of an <a href="/facts/Incidence_algebra/xrLTmg99">incidence algebra</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="relationship-to-the-dirac-delta-function">Relationship to the Dirac delta function</h2> <p>In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the Kronecker delta and <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> can both be used to represent a <a href="/facts/Discrete_distribution/EpsKKVRu">discrete distribution</a>. If the <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> of a distribution consists of points x = { x 1 , ⋯ , x n } {\displaystyle \mathbf {x} =\{x_{1},\cdots ,x_{n}\}} , with corresponding probabilities p 1 , ⋯ , p n {\displaystyle p_{1},\cdots ,p_{n}} , then the <a href="/facts/Probability_mass_function/LhurokRt">probability mass function</a> p ( x ) {\displaystyle p(x)} of the distribution over x {\displaystyle \mathbf {x} } can be written, using the Kronecker delta, as p ( x ) = ∑ i = 1 n p i δ x x i . {\displaystyle p(x)=\sum _{i=1}^{n}p_{i}\delta _{xx_{i}}.} </p><p>Equivalently, the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> f ( x ) {\displaystyle f(x)} of the distribution can be written using the Dirac delta function as f ( x ) = ∑ i = 1 n p i δ ( x − x i ) . {\displaystyle f(x)=\sum _{i=1}^{n}p_{i}\delta (x-x_{i}).} </p><p>Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function. For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the <a href="/facts/Nyquist%E2%80%93Shannon_sampling_theorem/JYlVrvc5">Nyquist–Shannon sampling theorem</a>, the resulting discrete-time signal will be a Kronecker delta function. </p> <h2 id="generalizations">Generalizations</h2> <p>If it is considered as a type ( 1 , 1 ) {\displaystyle (1,1)} <a href="/facts/Tensor/pNjFo5tV">tensor</a>, the Kronecker tensor can be written δ j i {\displaystyle \delta _{j}^{i}} with a <a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">covariant</a> index j {\displaystyle j} and <a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">contravariant</a> index i {\displaystyle i} : δ j i = { 0 ( i ≠ j ) , 1 ( i = j ) . {\displaystyle \delta _{j}^{i}={\begin{cases}0&(i\neq j),\\1&(i=j).\end{cases}}} </p><p>This tensor represents: </p> <ul><li>The identity mapping (or identity matrix), considered as a <a href="/facts/Linear_mapping/Q1PBKNVV">linear mapping</a> V → V {\displaystyle V\to V} or V ∗ → V ∗ {\displaystyle V^{*}\to V^{*}} </li> <li>The <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> or <a href="/facts/Tensor_contraction/fqz0zvWp">tensor contraction</a>, considered as a mapping V ∗ ⊗ V → K {\displaystyle V^{*}\otimes V\to K} </li> <li>The map K → V ∗ ⊗ V {\displaystyle K\to V^{*}\otimes V} , representing scalar multiplication as a sum of <a href="/facts/Outer_product/qR9C2BU1">outer products</a>.</li></ul> <p>The generalized Kronecker delta or multi-index Kronecker delta of order 2 p {\displaystyle 2p} is a type ( p , p ) {\displaystyle (p,p)} tensor that is completely <a href="/facts/Antisymmetric_tensor/atcSaPST">antisymmetric</a> in its p {\displaystyle p} upper indices, and also in its p {\displaystyle p} lower indices. </p><p>Two definitions that differ by a factor of p ! {\displaystyle p!} are in use. Below, the version is presented has nonzero components scaled to be ± 1 {\displaystyle \pm 1} . The second version has nonzero components that are ± 1 / p ! {\displaystyle \pm 1/p!} , with consequent changes scaling factors in formulae, such as the scaling factors of 1 / p ! {\displaystyle 1/p!} in <i>§ Properties of the generalized Kronecker delta</i> below disappearing.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h3>Definitions of the generalized Kronecker delta</h3> <p>In terms of the indices, the generalized Kronecker delta is defined as:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> δ ν 1 … ν p μ 1 … μ p = { − 1 if ν 1 … ν p are distinct integers and are an even permutation of μ 1 … μ p − 1 if ν 1 … ν p are distinct integers and are an odd permutation of μ 1 … μ p − 0 in all other cases . {\displaystyle \delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}={\begin{cases}{\phantom {-}}1&\quad {\text{if }}\nu _{1}\dots \nu _{p}{\text{ are distinct integers and are an even permutation of }}\mu _{1}\dots \mu _{p}\\-1&\quad {\text{if }}\nu _{1}\dots \nu _{p}{\text{ are distinct integers and are an odd permutation of }}\mu _{1}\dots \mu _{p}\\{\phantom {-}}0&\quad {\text{in all other cases}}.\end{cases}}} </p><p>Let S p {\displaystyle \mathrm {S} _{p}} be the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> of degree p {\displaystyle p} , then: δ ν 1 … ν p μ 1 … μ p = ∑ σ ∈ S p sgn ( σ ) δ ν σ ( 1 ) μ 1 ⋯ δ ν σ ( p ) μ p = ∑ σ ∈ S p sgn ( σ ) δ ν 1 μ σ ( 1 ) ⋯ δ ν p μ σ ( p ) . {\displaystyle \delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}=\sum _{\sigma \in \mathrm {S} _{p}}\operatorname {sgn}(\sigma )\,\delta _{\nu _{\sigma (1)}}^{\mu _{1}}\cdots \delta _{\nu _{\sigma (p)}}^{\mu _{p}}=\sum _{\sigma \in \mathrm {S} _{p}}\operatorname {sgn}(\sigma )\,\delta _{\nu _{1}}^{\mu _{\sigma (1)}}\cdots \delta _{\nu _{p}}^{\mu _{\sigma (p)}}.} </p><p>Using <a href="/facts/Antisymmetric_tensor/atcSaPST">anti-symmetrization</a>: δ ν 1 … ν p μ 1 … μ p = p ! δ [ ν 1 μ 1 … δ ν p ] μ p = p ! δ ν 1 [ μ 1 … δ ν p μ p ] . {\displaystyle \delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}=p!\delta _{[\nu _{1}}^{\mu _{1}}\dots \delta _{\nu _{p}]}^{\mu _{p}}=p!\delta _{\nu _{1}}^{[\mu _{1}}\dots \delta _{\nu _{p}}^{\mu _{p}]}.} </p><p>In terms of a p × p {\displaystyle p\times p} <a href="/facts/Determinant/eGYqJ2TF">determinant</a>:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> δ ν 1 … ν p μ 1 … μ p = | δ ν 1 μ 1 ⋯ δ ν p μ 1 ⋮ ⋱ ⋮ δ ν 1 μ p ⋯ δ ν p μ p | . {\displaystyle \delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}={\begin{vmatrix}\delta _{\nu _{1}}^{\mu _{1}}&\cdots &\delta _{\nu _{p}}^{\mu _{1}}\\\vdots &\ddots &\vdots \\\delta _{\nu _{1}}^{\mu _{p}}&\cdots &\delta _{\nu _{p}}^{\mu _{p}}\end{vmatrix}}.} </p><p>Using the <a href="/facts/Laplace_expansion/B4EIjlsp">Laplace expansion</a> (<a href="/facts/Determinant/eGYqJ2TF">Laplace's formula</a>) of determinant, it may be defined <a href="/facts/Recursion/CxSKxJoW">recursively</a>:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> δ ν 1 … ν p μ 1 … μ p = ∑ k = 1 p ( − 1 ) p + k δ ν k μ p δ ν 1 … ν ˇ k … ν p μ 1 … μ k … μ ˇ p = δ ν p μ p δ ν 1 … ν p − 1 μ 1 … μ p − 1 − ∑ k = 1 p − 1 δ ν k μ p δ ν 1 … ν k − 1 ν p ν k + 1 … ν p − 1 μ 1 … μ k − 1 μ k μ k + 1 … μ p − 1 , {\displaystyle {\begin{aligned}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}&=\sum _{k=1}^{p}(-1)^{p+k}\delta _{\nu _{k}}^{\mu _{p}}\delta _{\nu _{1}\dots {\check {\nu }}_{k}\dots \nu _{p}}^{\mu _{1}\dots \mu _{k}\dots {\check {\mu }}_{p}}\\&=\delta _{\nu _{p}}^{\mu _{p}}\delta _{\nu _{1}\dots \nu _{p-1}}^{\mu _{1}\dots \mu _{p-1}}-\sum _{k=1}^{p-1}\delta _{\nu _{k}}^{\mu _{p}}\delta _{\nu _{1}\dots \nu _{k-1}\,\nu _{p}\,\nu _{k+1}\dots \nu _{p-1}}^{\mu _{1}\dots \mu _{k-1}\,\mu _{k}\,\mu _{k+1}\dots \mu _{p-1}},\end{aligned}}} where the caron, ˇ {\displaystyle {\check {}}} , indicates an index that is omitted from the sequence. </p><p>When p = n {\displaystyle p=n} (the dimension of the vector space), in terms of the <a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a>: δ ν 1 … ν n μ 1 … μ n = ε μ 1 … μ n ε ν 1 … ν n . {\displaystyle \delta _{\nu _{1}\dots \nu _{n}}^{\mu _{1}\dots \mu _{n}}=\varepsilon ^{\mu _{1}\dots \mu _{n}}\varepsilon _{\nu _{1}\dots \nu _{n}}\,.} More generally, for m = n − p {\displaystyle m=n-p} , using the <a href="/facts/Einstein_summation_convention/UXwS43wU">Einstein summation convention</a>: δ ν 1 … ν p μ 1 … μ p = 1 m ! ε κ 1 … κ m μ 1 … μ p ε κ 1 … κ m ν 1 … ν p . {\displaystyle \delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}={\tfrac {1}{m!}}\varepsilon ^{\kappa _{1}\dots \kappa _{m}\mu _{1}\dots \mu _{p}}\varepsilon _{\kappa _{1}\dots \kappa _{m}\nu _{1}\dots \nu _{p}}\,.} </p> <h3>Contractions of the generalized Kronecker delta</h3> <p>Kronecker Delta contractions depend on the dimension of the space. For example, δ μ 1 ν 1 δ ν 1 ν 2 μ 1 μ 2 = ( d − 1 ) δ ν 2 μ 2 , {\displaystyle \delta _{\mu _{1}}^{\nu _{1}}\delta _{\nu _{1}\nu _{2}}^{\mu _{1}\mu _{2}}=(d-1)\delta _{\nu _{2}}^{\mu _{2}},} where d is the dimension of the space. From this relation the full contracted delta is obtained as δ μ 1 μ 2 ν 1 ν 2 δ ν 1 ν 2 μ 1 μ 2 = 2 d ( d − 1 ) . {\displaystyle \delta _{\mu _{1}\mu _{2}}^{\nu _{1}\nu _{2}}\delta _{\nu _{1}\nu _{2}}^{\mu _{1}\mu _{2}}=2d(d-1).} The generalization of the preceding formulas is δ μ 1 … μ n ν 1 … ν n δ ν 1 … ν p μ 1 … μ p = n ! ( d − p + n ) ! ( d − p ) ! δ ν n + 1 … ν p μ n + 1 … μ p . {\displaystyle \delta _{\mu _{1}\dots \mu _{n}}^{\nu _{1}\dots \nu _{n}}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}=n!{\frac {(d-p+n)!}{(d-p)!}}\delta _{\nu _{n+1}\dots \nu _{p}}^{\mu _{n+1}\dots \mu _{p}}.} </p> <h3>Properties of the generalized Kronecker delta</h3> <p>The generalized Kronecker delta may be used for <a href="/facts/Antisymmetric_tensor/atcSaPST">anti-symmetrization</a>: 1 p ! δ ν 1 … ν p μ 1 … μ p a ν 1 … ν p = a [ μ 1 … μ p ] , 1 p ! δ ν 1 … ν p μ 1 … μ p a μ 1 … μ p = a [ ν 1 … ν p ] . {\displaystyle {\begin{aligned}{\frac {1}{p!}}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}a^{\nu _{1}\dots \nu _{p}}&=a^{[\mu _{1}\dots \mu _{p}]},\\{\frac {1}{p!}}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}a_{\mu _{1}\dots \mu _{p}}&=a_{[\nu _{1}\dots \nu _{p}]}.\end{aligned}}} </p><p>From the above equations and the properties of <a href="/facts/Anti-symmetric_tensor/atcSaPST">anti-symmetric tensors</a>, we can derive the properties of the generalized Kronecker delta: 1 p ! δ ν 1 … ν p μ 1 … μ p a [ ν 1 … ν p ] = a [ μ 1 … μ p ] , 1 p ! δ ν 1 … ν p μ 1 … μ p a [ μ 1 … μ p ] = a [ ν 1 … ν p ] , 1 p ! δ ν 1 … ν p μ 1 … μ p δ κ 1 … κ p ν 1 … ν p = δ κ 1 … κ p μ 1 … μ p , {\displaystyle {\begin{aligned}{\frac {1}{p!}}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}a^{[\nu _{1}\dots \nu _{p}]}&=a^{[\mu _{1}\dots \mu _{p}]},\\{\frac {1}{p!}}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}a_{[\mu _{1}\dots \mu _{p}]}&=a_{[\nu _{1}\dots \nu _{p}]},\\{\frac {1}{p!}}\delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}\delta _{\kappa _{1}\dots \kappa _{p}}^{\nu _{1}\dots \nu _{p}}&=\delta _{\kappa _{1}\dots \kappa _{p}}^{\mu _{1}\dots \mu _{p}},\end{aligned}}} which are the generalized version of formulae written in <i>§ Properties</i>. The last formula is equivalent to the <a href="/facts/Cauchy%E2%80%93Binet_formula/jbRCH0tj">Cauchy–Binet formula</a>. </p><p>Reducing the order via summation of the indices may be expressed by the identity<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> δ ν 1 … ν s μ s + 1 … μ p μ 1 … μ s μ s + 1 … μ p = ( n − s ) ! ( n − p ) ! δ ν 1 … ν s μ 1 … μ s . {\displaystyle \delta _{\nu _{1}\dots \nu _{s}\,\mu _{s+1}\dots \mu _{p}}^{\mu _{1}\dots \mu _{s}\,\mu _{s+1}\dots \mu _{p}}={\frac {(n-s)!}{(n-p)!}}\delta _{\nu _{1}\dots \nu _{s}}^{\mu _{1}\dots \mu _{s}}.} </p><p>Using both the summation rule for the case p = n {\displaystyle p=n} and the relation with the Levi-Civita symbol, <a href="/facts/Levi-Civita_symbol/WBKUA3Tw">the summation rule of the Levi-Civita symbol</a> is derived: δ ν 1 … ν p μ 1 … μ p = 1 ( n − p ) ! ε μ 1 … μ p κ p + 1 … κ n ε ν 1 … ν p κ p + 1 … κ n . {\displaystyle \delta _{\nu _{1}\dots \nu _{p}}^{\mu _{1}\dots \mu _{p}}={\frac {1}{(n-p)!}}\varepsilon ^{\mu _{1}\dots \mu _{p}\,\kappa _{p+1}\dots \kappa _{n}}\varepsilon _{\nu _{1}\dots \nu _{p}\,\kappa _{p+1}\dots \kappa _{n}}.} The 4D version of the last relation appears in Penrose's <a href="/facts/Mathematics_of_general_relativity/5RktLkom">spinor approach to general relativity</a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> that he later generalized, while he was developing Aitken's diagrams,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> to become part of the technique of <a href="/facts/Penrose_graphical_notation/VUA7KLpm">Penrose graphical notation</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Also, this relation is extensively used in <a href="/facts/S-duality/eyWposrn">S-duality</a> theories, especially when written in the language of <a href="/facts/Differential_form/T9gyHtMv">differential forms</a> and <a href="/facts/Hodge_star_operator/AH6tmHhU">Hodge duals</a>. </p> <h2 id="integral-representations">Integral representations</h2> <p>For any integers j {\displaystyle j} and k {\displaystyle k} , the Kronecker delta can be written as a complex <a href="/facts/Contour_integral/4ILprcRr">contour integral</a> using a standard <a href="/facts/Residue_(complex_analysis)/GllQJk32">residue</a> calculation. The integral is taken over the unit circle in the complex plane, oriented counterclockwise. An equivalent representation of the integral arises by parameterizing the contour by an angle around the origin. δ j k = 1 2 π i ∮ | z | = 1 z j − k − 1 d z = 1 2 π ∫ 0 2 π e i ( j − k ) φ d φ {\displaystyle \delta _{jk}={\frac {1}{2\pi i}}\oint _{|z|=1}z^{j-k-1}\,dz={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{i(j-k)\varphi }\,d\varphi } </p> <h2 id="the-kronecker-comb">The Kronecker comb</h2> <p>The Kronecker comb function with period N {\displaystyle N} is defined (using <a href="/facts/Digital_signal_processing/PEG7azr8">DSP</a> notation) as: Δ N [ n ] = ∑ k = − ∞ ∞ δ [ n − k N ] , {\displaystyle \Delta _{N}[n]=\sum _{k=-\infty }^{\infty }\delta [n-kN],} where N ≠ 0 {\displaystyle N\neq 0} and n {\displaystyle n} are integers. The Kronecker comb thus consists of an infinite series of unit impulses that are N units apart, aligned so one of the impulses occurs at zero. It may be considered to be the discrete analog of the <a href="/facts/Dirac_comb/0tx3xslS">Dirac comb</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dirac_measure/mj8Qfhr2">Dirac measure</a></li> <li><a href="/facts/Indicator_function/QEuh04NM">Indicator function</a></li> <li><a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a></li> <li><a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a></li> <li><a href="/facts/Minkowski_space/vDkfkQJn">Minkowski metric</a></li> <li><a href="/facts/%27t_Hooft_symbol/tzAnv0a2">'t Hooft symbol</a></li> <li><a href="/facts/Unit_function/YFHumrlC">Unit function</a></li> <li><a href="/facts/XNOR_gate/LW793PwY">XNOR gate</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Trowbridge, J. H. (1998). "On a Technique for Measurement of Turbulent Shear Stress in the Presence of Surface Waves". Journal of Atmospheric and Oceanic Technology. 15 (1): 291. Bibcode:1998JAtOT..15..290T. doi:10.1175/1520-0426(1998)015<0290:OATFMO>2.0.CO;2. <a href="https://doi.org/10.1175%2F1520-0426%281998%29015%3C0290%3AOATFMO%3E2.0.CO%3B2" target="_blank">https://doi.org/10.1175%2F1520-0426%281998%29015%3C0290%3AOATFMO%3E2.0.CO%3B2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dirac, Paul (1930). The Principles of Quantum Mechanics (1st ed.). Oxford University Press. ISBN 9780198520115. <a href="9780198520115" target="_blank">9780198520115</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Spiegel, Eugene; O'Donnell, Christopher J. (1997), Incidence Algebras, Pure and Applied Mathematics, vol. 206, Marcel Dekker, ISBN 0-8247-0036-8. <a href="0-8247-0036-8" target="_blank">0-8247-0036-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pope, Christopher (2008). "Geometry and Group Theory" (PDF). <a href="http://people.physics.tamu.edu/pope/geom-group.pdf" target="_blank">http://people.physics.tamu.edu/pope/geom-group.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Frankel, Theodore (2012). The Geometry of Physics: An Introduction (3rd ed.). Cambridge University Press. ISBN 9781107602601. <a href="9781107602601" target="_blank">9781107602601</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Agarwal, D. C. (2007). Tensor Calculus and Riemannian Geometry (22nd ed.). Krishna Prakashan Media.[ISBN missing] <a href="/wiki/Wikipedia:Citing_sources" target="_blank">/wiki/Wikipedia:Citing_sources</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Lovelock, David; Rund, Hanno (1989). Tensors, Differential Forms, and Variational Principles. Courier Dover Publications. ISBN 0-486-65840-6. <a href="0-486-65840-6" target="_blank">0-486-65840-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>A recursive definition requires a first case, which may be taken as δ = 1 for p = 0, or alternatively δμν = δμν for p = 1 (generalized delta in terms of standard delta). <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Hassani, Sadri (2008). Mathematical Methods: For Students of Physics and Related Fields (2nd ed.). Springer-Verlag. ISBN 978-0-387-09503-5. <a href="978-0-387-09503-5" target="_blank">978-0-387-09503-5</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Penrose, Roger (June 1960). "A spinor approach to general relativity". Annals of Physics. 10 (2): 171–201. Bibcode:1960AnPhy..10..171P. doi:10.1016/0003-4916(60)90021-X. <a href="https://linkinghub.elsevier.com/retrieve/pii/000349166090021X" target="_blank">https://linkinghub.elsevier.com/retrieve/pii/000349166090021X</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Aitken, Alexander Craig (1958). Determinants and Matrices. UK: Oliver and Boyd. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). <a href="/wiki/Roger_Penrose" target="_blank">/wiki/Roger_Penrose</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> </ol>