# relations to Jacobian elliptic functions

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##### 1: 22.2 Definitions
22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$
##### 3: 20.9 Relations to Other Functions
###### §20.9(ii) EllipticFunctions and Modular Functions
See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. …
##### 7: Bille C. Carlson
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. …
##### 8: 22.17 Moduli Outside the Interval [0,1]
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. …
##### 10: 29.2 Differential Equations
For $\operatorname{sn}\left(z,k\right)$ see §22.2. …
###### §29.2(ii) Other Forms
For $\operatorname{am}\left(z,k\right)$ see §22.16(i). … we have …For the Weierstrass function $\wp$ see §23.2(ii). …