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20 Theta FunctionsProperties

§20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

With k defined by

20.9.1 k=θ22(0|τ)/θ32(0|τ)

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

20.9.2 K(k) =12πθ32(0|τ),
K(k) =-iτK(k),

together with (22.2.1).

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

20.9.3 RF(θ22(z,q)θ22(0,q),θ32(z,q)θ32(0,q),θ42(z,q)θ42(0,q))=θ1(0,q)θ1(z,q)z,
20.9.4 RF(0,θ34(0,q),θ44(0,q))=12π,
20.9.5 exp(-πRF(0,k2,1)RF(0,k2,1))=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 τ (or q) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485).

As a function of τ, k2 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.