# §20.9(i) Elliptic Integrals

With $k$ defined by

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

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

 20.9.2 $\displaystyle\mathop{K\/}\nolimits\!\left(k\right)$ $\displaystyle=\tfrac{1}{2}\pi{\mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0% \middle|\tau\right),$ $\displaystyle{\mathop{K\/}\nolimits^{\prime}}\!\left(k\right)$ $\displaystyle=-i\tau\mathop{K\/}\nolimits\!\left(k\right),$

together with (22.2.1).

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

 20.9.3 $\mathop{R_{F}\/}\nolimits\!\left(\frac{{\mathop{\theta_{2}\/}\nolimits^{2}}\!% \left(z,q\right)}{{\mathop{\theta_{2}\/}\nolimits^{2}}\!\left(0,q\right)},% \frac{{\mathop{\theta_{3}\/}\nolimits^{2}}\!\left(z,q\right)}{{\mathop{\theta_% {3}\/}\nolimits^{2}}\!\left(0,q\right)},\frac{{\mathop{\theta_{4}\/}\nolimits^% {2}}\!\left(z,q\right)}{{\mathop{\theta_{4}\/}\nolimits^{2}}\!\left(0,q\right)% }\right)=\frac{{\mathop{\theta_{1}\/}\nolimits^{\prime}}\!\left(0,q\right)}{% \mathop{\theta_{1}\/}\nolimits\!\left(z,q\right)}z,$
 20.9.4 $\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\theta_{3}\/}\nolimits^{4}}\!\left% (0,q\right),{\mathop{\theta_{4}\/}\nolimits^{4}}\!\left(0,q\right)\right)=% \tfrac{1}{2}\pi,$
 20.9.5 $\mathop{\exp\/}\nolimits\!\left(-\frac{\pi\mathop{R_{F}\/}\nolimits\!\left(0,k% ^{2},1\right)}{\mathop{R_{F}\/}\nolimits\!\left(0,{k^{\prime}}^{2},1\right)}% \right)=q.$

# §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.