# Jacobian elliptic functions

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##### 1: 22.15 Inverse Functions
###### §22.15(i) Definitions
The principal values satisfy …
##### 2: 22.2 Definitions
###### §22.2 Definitions
22.2.4 $\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},$
22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$
As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … …
##### 3: 22.17 Moduli Outside the Interval [0,1]
###### §22.17(ii) Complex Moduli
When $z$ is fixed each of the twelve Jacobian elliptic functions is a meromorphic function of $k^{2}$. …For proofs of these results and further information see Walker (2003).
##### 4: 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 $\mathrm{Z}\left(x|k\right)$. … 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). …
See §22.17.
##### 7: 22.4 Periods, Poles, and Zeros
###### §22.4(ii) Graphical Interpretation via Glaisher’s Notation
22.14.1 $\int\operatorname{sn}\left(x,k\right)\,\mathrm{d}x=k^{-1}\ln\left(% \operatorname{dn}\left(x,k\right)-k\operatorname{cn}\left(x,k\right)\right),$