# Jacobian elliptic-function form

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##### 1: 22.15 Inverse Functions
###### §22.15(ii) Representations as Elliptic Integrals
The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. …
##### 2: 22.2 Definitions
###### §22.2 Definitions
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. … … The Jacobian functions are related in the following way. …
##### 3: 22.17 Moduli Outside the Interval [0,1]
###### §22.17(i) Real or Purely Imaginary Moduli
Jacobian elliptic functions with real moduli in the intervals $(-\infty,0)$ and $(1,\infty)$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. …
###### §22.17(ii) Complex Moduli
For proofs of these results and further information see Walker (2003).
##### 4: 22.5 Special Values
###### §22.5 Special Values
Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $iK^{\prime}$. …
##### 5: 22.16 Related Functions
For $-K, …See Figure 22.16.2. …
22.16.18 $\mathcal{E}\left(x,k\right)=-k^{2}\int_{0}^{x}{\operatorname{cd}}^{2}\left(t,k% \right)\mathrm{d}t+x+k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}% \left(x,k\right),$
22.16.21 $\mathcal{E}\left(x,k\right)=-\int_{0}^{x}{\operatorname{dc}}^{2}\left(t,k% \right)\mathrm{d}t+x+\operatorname{sn}\left(x,k\right)\operatorname{dc}\left(x% ,k\right),$
22.16.26 $\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({\operatorname{cs}}^{2}\left(t,% k\right)-t^{-2}\right)\mathrm{d}t+x^{-1}-\operatorname{cn}\left(x,k\right)% \operatorname{ds}\left(x,k\right).$
##### 6: 22.18 Mathematical Applications
###### §22.18(i) Lengths and Parametrization of Plane Curves
The special case $y^{2}=(1-x^{2})(1-k^{2}x^{2})$ is in Jacobian normal form. …
##### 7: 22.14 Integrals
###### §22.14(ii) Indefinite Integrals of Powers of JacobianEllipticFunctions
The indefinite integral of the 3rd power of a Jacobian function can be expressed as an elementary function of Jacobian functions and a product of Jacobian functions. …
##### 8: 22.4 Periods, Poles, and Zeros
###### §22.4(i) Distribution
For the distribution of the $k$-zeros of the Jacobian elliptic functions see Walker (2009).
###### §22.4(ii) Graphical Interpretation via Glaisher’s Notation
22.13.4 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cd}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{cd}}^{2}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{cd}}^{2}\left(z,k\right)\right),$