# Jacobi function

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##### 2: 22.6 Elementary Identities
22.6.2 $1+{\operatorname{cs}}^{2}\left(z,k\right)=k^{2}+{\operatorname{ds}}^{2}\left(z% ,k\right)={\operatorname{ns}}^{2}\left(z,k\right),$
22.6.5 $\operatorname{sn}\left(2z,k\right)=\frac{2\operatorname{sn}\left(z,k\right)% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)}{1-k^{2}{% \operatorname{sn}}^{4}\left(z,k\right)},$
22.6.14 $\operatorname{ns}\left(2z,k\right)=\frac{{\operatorname{ns}}^{4}\left(z,k% \right)-k^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k% \right)\operatorname{ns}\left(z,k\right)},$
##### 3: 35.7 Gaussian Hypergeometric Function of Matrix Argument
###### Jacobi Form
35.7.2 $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$; $\gamma,\delta,\nu\in\mathbb{C}$; $\Re\left(\gamma\right)>-1$.
22.8.3 $\operatorname{dn}(u+v)=\frac{\operatorname{dn}u\operatorname{dn}v-k^{2}% \operatorname{sn}u\operatorname{cn}u\operatorname{sn}v\operatorname{cn}v}{1-k^% {2}{\operatorname{sn}}^{2}u{\operatorname{sn}}^{2}v}.$
22.8.5 $\operatorname{sd}(u+v)=\frac{\operatorname{sd}u\operatorname{cd}v\operatorname% {nd}v+\operatorname{sd}v\operatorname{cd}u\operatorname{nd}u}{1+k^{2}{k^{% \prime}}^{2}{\operatorname{sd}}^{2}u{\operatorname{sd}}^{2}v},$
22.8.8 $\operatorname{nc}(u+v)=\frac{\operatorname{nc}u\operatorname{nc}v+% \operatorname{sc}u\operatorname{dc}u\operatorname{sc}v\operatorname{dc}v}{1-{k% ^{\prime}}^{2}{\operatorname{sc}}^{2}u{\operatorname{sc}}^{2}v},$
##### 5: 14.31 Other Applications
###### §14.31(ii) Conical Functions
The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
##### 6: 15.17 Mathematical Applications
###### §15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …
##### 7: 22.14 Integrals
22.14.1 $\int\operatorname{sn}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{dn}\left(x,k\right)-k\operatorname{cn}\left(x,k\right)\right),$
22.14.3 $\int\operatorname{dn}\left(x,k\right)\,\mathrm{d}x=\operatorname{Arcsin}\left(% \operatorname{sn}\left(x,k\right)\right)=\operatorname{am}\left(x,k\right).$
22.14.4 $\int\operatorname{cd}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{nd}\left(x,k\right)+k\operatorname{sd}\left(x,k\right)\right),$
22.14.7 $\int\operatorname{dc}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{nc}% \left(x,k\right)+\operatorname{sc}\left(x,k\right)\right),$
22.14.10 $\int\operatorname{ns}\left(x,k\right)\,\mathrm{d}x=\ln\left(\operatorname{ds}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),$
##### 8: 22.4 Periods, Poles, and Zeros
For example, the poles of $\operatorname{sn}\left(z,k\right)$, abbreviated as $\operatorname{sn}$ in the following tables, are at $z=2mK+(2n+1)iK^{\prime}$. … Then: (a) In any lattice unit cell $\operatorname{pq}\left(z,k\right)$ has a simple zero at $z=\mbox{p}$ and a simple pole at $z=\mbox{q}$. (b) The difference between p and the nearest q is a half-period of $\operatorname{pq}\left(z,k\right)$. … For example, $\operatorname{sn}\left(z+K,k\right)=\operatorname{cd}\left(z,k\right)$. …
##### 9: 22.5 Special Values
For example, at $z=K+iK^{\prime}$, $\operatorname{sn}\left(z,k\right)=1/k$, $\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0$. … Table 22.5.2 gives $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$ for other special values of $z$. For example, $\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}$. … In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
##### 10: 22.15 Inverse Functions
22.15.1 $\operatorname{sn}\left(\xi,k\right)=x,$ $-1\leq x\leq 1$,
22.15.2 $\operatorname{cn}\left(\eta,k\right)=x,$ $-1\leq x\leq 1$,
are denoted respectively by …
22.15.11 $x=\int_{0}^{\operatorname{sn}\left(x,k\right)}\frac{\,\mathrm{d}t}{\sqrt{(1-t^% {2})(1-k^{2}t^{2})}},$ $-1\leq x\leq 1$, $0\leq k\leq 1$.