§20.9 Relations to Other Functions

With $k$ defined by

20.9.1		$$k={\theta }_2^2\left(0\right|\tau )/{\theta }_3^2\left(0\right|\tau )$$
ⓘ Defines: $k$: modulus (locally) Symbols: ${\theta }_j\left(z\right|\tau )$: 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 Ch.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		$K\left(k\right)$	$=\frac{1}{2}\pi {\theta }_3^2\left(0\right|\tau ),$
		$K^{\prime }\left(k\right)$	$=-\mathrm{i}\tau K\left(k\right),$
ⓘ Symbols: ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tau$: lattice parameter and $k$: modulus Referenced by: §20.11(iii), §20.11(iii), §20.9(i), §20.9(ii), §23.6(ii) Permalink: http://dlmf.nist.gov/20.9.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §20.9(i), §20.9 and Ch.20

together with (22.2.1).

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

20.9.3		$$R_F(\frac{{\theta }_2^2(z,q)}{{\theta }_2^2(0,q)},\frac{{\theta }_3^2(z,q)}{{\theta }_3^2(0,q)},\frac{{\theta }_4^2(z,q)}{{\theta }_4^2(0,q)})=\frac{{\theta }_1^{\prime }(0,q)}{{\theta }_1(z,q)}z,$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: 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 Ch.20

20.9.4		$$R_F(0,{\theta }_3^4(0,q),{\theta }_4^4(0,q))=\frac{1}{2}\pi ,$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter and $q$: nome Referenced by: §20.9(i) Permalink: http://dlmf.nist.gov/20.9.E4 Encodings: TeX, pMML, png See also: Annotations for §20.9(i), §20.9 and Ch.20

20.9.5		$$\exp \left(-\frac{\pi R_F(0,k^2,1)}{R_F(0,k^{\prime }{}^2,1)}\right)=q.$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $q$: nome and $k$: modulus Referenced by: §20.9(i) Permalink: http://dlmf.nist.gov/20.9.E5 Encodings: TeX, pMML, png See also: Annotations for §20.9(i), §20.9 and Ch.20

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).

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