# §20.9 Relations to Other Functions

## §20.9(i) Elliptic Integrals

With $k$ defined by

 20.9.1 $k={\theta_{2}^{2}}\left(0\middle|\tau\right)/{\theta_{3}^{2}}\left(0\middle|% \tau\right)$ ⓘ Defines: $k$: modulus (locally) Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function and $\tau$: lattice parameter Referenced by: §20.11(iii), §20.9(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/20.9.E1 Encodings: TeX, pMML, png See also: Annotations for 20.9(i), 20.9 and 20

and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions:

 20.9.2 $\displaystyle K\left(k\right)$ $\displaystyle=\tfrac{1}{2}\pi{\theta_{3}^{2}}\left(0\middle|\tau\right),$ $\displaystyle K'\left(k\right)$ $\displaystyle=-i\tau K\left(k\right),$

together with (22.2.1).

In the case of the symmetric integrals, with the notation of §19.16(i) we have

 20.9.3 $R_{F}\left(\frac{{\theta_{2}^{2}}\left(z,q\right)}{{\theta_{2}^{2}}\left(0,q% \right)},\frac{{\theta_{3}^{2}}\left(z,q\right)}{{\theta_{3}^{2}}\left(0,q% \right)},\frac{{\theta_{4}^{2}}\left(z,q\right)}{{\theta_{4}^{2}}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,$ ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $z$: complex and $q$: nome Referenced by: §19.25(v), §20.9(i) Permalink: http://dlmf.nist.gov/20.9.E3 Encodings: TeX, pMML, png See also: Annotations for 20.9(i), 20.9 and 20
 20.9.4 $R_{F}\left(0,{\theta_{3}^{4}}\left(0,q\right),{\theta_{4}^{4}}\left(0,q\right)% \right)=\tfrac{1}{2}\pi,$
 20.9.5 $\exp\left(-\frac{\pi R_{F}\left(0,k^{2},1\right)}{R_{F}\left(0,{k^{\prime}}^{2% },1\right)}\right)=q.$

## §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).

As a function of $\tau$, $k^{2}$ is the elliptic modular function; see Walker (1996, Chapter 7) and (23.15.2), (23.15.6).

## §20.9(iii) Riemann Zeta Function

See Koblitz (1993, Ch. 2, §4) and Titchmarsh (1986b, pp. 21–22). See also §§20.10(i) and 25.2.