# 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: 1.13 Differential Equations
###### §1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
This is the Sturm-Liouville form of a second order differential equation, where denotes $\frac{\mathrm{d}}{\mathrm{d}x}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form
##### 4: 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. …
##### 5: 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).
##### 7: 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 $K$, $iK^{\prime}$. …
##### 8: Bille C. Carlson
In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …
##### 9: 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).