# Jacobi elliptic

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##### 1: 22.16 Related Functions
###### Special Values
22.8.10 $\operatorname{ns}(u+v)=\frac{\operatorname{ns}u\operatorname{ds}v\operatorname% {cs}v-\operatorname{ns}v\operatorname{ds}u\operatorname{cs}u}{{\operatorname{% cs}^{2}}v-{\operatorname{cs}^{2}}u},$
22.8.11 $\operatorname{ds}(u+v)=\frac{\operatorname{ds}u\operatorname{cs}v\operatorname% {ns}v-\operatorname{ds}v\operatorname{cs}u\operatorname{ns}u}{{\operatorname{% cs}^{2}}v-{\operatorname{cs}^{2}}u},$
22.8.12 $\operatorname{cs}(u+v)=\frac{\operatorname{cs}u\operatorname{ds}v\operatorname% {ns}v-\operatorname{cs}v\operatorname{ds}u\operatorname{ns}u}{{\operatorname{% cs}^{2}}v-{\operatorname{cs}^{2}}u}.$
22.8.14 $\operatorname{sn}(u+v)=\frac{\operatorname{sn}u\operatorname{cn}u\operatorname% {dn}v+\operatorname{sn}v\operatorname{cn}v\operatorname{dn}u}{\operatorname{cn% }u\operatorname{cn}v+\operatorname{sn}u\operatorname{dn}u\operatorname{sn}v% \operatorname{dn}v},$
22.8.23 $\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.$
##### 3: 22.6 Elementary Identities
22.6.2 $1+{\operatorname{cs}^{2}}\left(z,k\right)=k^{2}+{\operatorname{ds}^{2}}\left(z% ,k\right)={\operatorname{ns}^{2}}\left(z,k\right),$
22.6.14 $\operatorname{ns}\left(2z,k\right)=\frac{{\operatorname{ns}^{4}}\left(z,k% \right)-k^{2}}{2\operatorname{cs}\left(z,k\right)\operatorname{ds}\left(z,k% \right)\operatorname{ns}\left(z,k\right)},$
##### 6: 22.21 Tables
###### §22.21 Tables
Spenceley and Spenceley (1947) tabulates $\operatorname{sn}\left(Kx,k\right)$, $\operatorname{cn}\left(Kx,k\right)$, $\operatorname{dn}\left(Kx,k\right)$, $\operatorname{am}\left(Kx,k\right)$, $\mathcal{E}\left(Kx,k\right)$ for $\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}$ and $x=0\left(\tfrac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\operatorname{am}\left(Kx,k\right)$. Curtis (1964b) tabulates $\operatorname{sn}\left(mK/n,k\right)$, $\operatorname{cn}\left(mK/n,k\right)$, $\operatorname{dn}\left(mK/n,k\right)$ for $n=2(1)15$, $m=1(1)n-1$, and $q$ (not $k$) $=0(.005)0.35$ to 20D. Lawden (1989, pp. 280–284 and 293–297) tabulates $\operatorname{sn}\left(x,k\right)$, $\operatorname{cn}\left(x,k\right)$, $\operatorname{dn}\left(x,k\right)$, $\mathcal{E}\left(x,k\right)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. … Zhang and Jin (1996, p. 678) tabulates $\operatorname{sn}\left(Kx,k\right)$, $\operatorname{cn}\left(Kx,k\right)$, $\operatorname{dn}\left(Kx,k\right)$ for $k=\frac{1}{4},\frac{1}{2}$ and $x=0(.1)4$ to 7D. …
##### 7: 22.7 Landen Transformations
22.7.2 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}^{2}}\left(z/(1+k_{1}),k_{1}% \right)},$
22.7.3 $\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}^{2}}\left(z/(1+k_{1}),k_{1}\right)},$
22.7.4 $\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}^{2}}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}^{2}}\left(z/(1+k_{1}),k_% {1}\right)}.$
22.7.6 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},$
22.7.8 $\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}^% {2}}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$
##### 8: 22.14 Integrals
See §22.16(i) for $\operatorname{am}\left(z,k\right)$. …
22.14.11 $\int\operatorname{ds}\left(x,k\right)\mathrm{d}x=\ln\left(\operatorname{ns}% \left(x,k\right)-\operatorname{cs}\left(x,k\right)\right),$
22.14.13 $\int\frac{\mathrm{d}x}{\operatorname{sn}\left(x,k\right)}=\ln\left(\frac{% \operatorname{sn}\left(x,k\right)}{\operatorname{cn}\left(x,k\right)+% \operatorname{dn}\left(x,k\right)}\right),$
22.14.14 $\int\frac{\operatorname{cn}\left(x,k\right)\mathrm{d}x}{\operatorname{sn}\left% (x,k\right)}=\frac{1}{2}\ln\left(\frac{1-\operatorname{dn}\left(x,k\right)}{1+% \operatorname{dn}\left(x,k\right)}\right),$
22.14.15 $\int\frac{\operatorname{cn}\left(x,k\right)\mathrm{d}x}{{\operatorname{sn}^{2}% }\left(x,k\right)}=-\frac{\operatorname{dn}\left(x,k\right)}{\operatorname{sn}% \left(x,k\right)}.$
##### 9: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$; the nine subsidiary Jacobian elliptic functions $\operatorname{cd}\left(z,k\right)$, $\operatorname{sd}\left(z,k\right)$, $\operatorname{nd}\left(z,k\right)$, $\operatorname{dc}\left(z,k\right)$, $\operatorname{nc}\left(z,k\right)$, $\operatorname{sc}\left(z,k\right)$, $\operatorname{ns}\left(z,k\right)$, $\operatorname{ds}\left(z,k\right)$, $\operatorname{cs}\left(z,k\right)$; the amplitude function $\operatorname{am}\left(x,k\right)$; Jacobi’s epsilon and zeta functions $\mathcal{E}\left(x,k\right)$ and $\mathrm{Z}\left(x|k\right)$. … The notation $\operatorname{sn}\left(z,k\right)$, $\operatorname{cn}\left(z,k\right)$, $\operatorname{dn}\left(z,k\right)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\operatorname{sn}\left(z,k\right)$ are $\mathrm{sn}(z\mathpunct{|}m)$ and $\mathrm{sn}(z,m)$ with $m=k^{2}$; see Abramowitz and Stegun (1964) and Walker (1996). …
##### 10: 22.17 Moduli Outside the Interval [0,1]
22.17.1 $\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),$
22.17.2 $\operatorname{sn}\left(z,1/k\right)=k\operatorname{sn}\left(z/k,k\right),$
22.17.7 $\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),$