# §20.9 Relations to Other Functions

## §20.9(i) Elliptic Integrals

With defined by

20.9.1

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

together with (22.2.1).

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

## §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 and (or ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485).

As a function of , 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.