# relations to theta functions

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##### 1: 20.9 Relations to Other Functions
###### §20.9(ii) Elliptic Functions and Modular Functions
See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …
##### 3: 22.2 Definitions
22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$
##### 6: 19.10 Relations to Other Functions
###### §19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
##### 8: 19.25 Relations to Other Functions
###### §19.25(iv) ThetaFunctions
For relations of symmetric integrals to theta functions, see §20.9(i). …
##### 10: 20.11 Generalizations and Analogs
However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau=\tau(k)$ of the theta functions via (20.9.2). …