20 Theta Functions Properties 20.8 Watson’s Expansions 20.10 Integrals

§20.9 Relations to Other Functions

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Permalink:: http://dlmf.nist.gov/20.9
See also:: Annotations for Ch.20

§20.9(i) Elliptic Integrals
§20.9(ii) Elliptic Functions and Modular Functions
§20.9(iii) Riemann Zeta Function

§20.9(i) Elliptic Integrals

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Keywords:: elliptic integrals, relations to other functions, theta functions
Notes:: See Walker (1996, p. 156), Whittaker and Watson (1927, p. 481), Serre (1973, p. 109), and McKean and Moll (1999, §3.3). For (20.9.3) combination of (20.4.6) and (23.6.5) – (23.6.7) yields $\wp\left(z\right)-e_{j}=\left(\frac{v\theta_{1}'\left(0,q\right)\theta_{j+1}% \left(v,q\right)}{z\theta_{1}\left(v,q\right)\theta_{j+1}\left(0,q\right)}% \right)^{2}$ , $j=1,2,3$ , where $v=\pi z/(2\omega_{1})$ . Then by application of (19.25.35) and use of the properties that $R_{F}$ is homogenous and of degree $-\tfrac{1}{2}$ in its three variables (§§19.16(ii), 19.16(iii)), we derive $z=\frac{z\theta_{1}\left(v,q\right)}{v\theta_{1}'\left(0,q\right)}R_{F}\left(% \frac{{\theta_{2}}^{2}\left(v,q\right)}{{\theta_{2}}^{2}\left(0,q\right)},% \frac{{\theta_{3}}^{2}\left(v,q\right)}{{\theta_{3}}^{2}\left(0,q\right)},% \frac{{\theta_{4}}^{2}\left(v,q\right)}{{\theta_{4}}^{2}\left(0,q\right)}\right)$ . This equation becomes (20.9.3) when the $z$ ’s are cancelled and $v$ is renamed $z$ . For (20.9.4), from (19.25.1) and Erdélyi et al. (1953b, 13.20(11)) we have $K\left(k\right)=R_{F}\left(0,\frac{{\theta_{4}}^{4}\left(0,q\right)}{{\theta_{% 3}}^{4}\left(0,q\right)},1\right)={\theta_{3}}^{2}\left(0,q\right)R_{F}\left(0% ,{\theta_{3}}^{4}\left(0,q\right),{\theta_{4}}^{4}\left(0,q\right)\right)$ , where the second equality uses the homogeneity and symmetry of $R_{F}$ . Comparison with (20.9.2) proves (20.9.4). For (20.9.5), by (19.25.1) the left side is $\exp\left(-\pi K\left(k^{\prime}\right)/K\left(k\right)\right)$ , which equals $q$ by Erdélyi et al. (1953b, 13.19(4)).
Referenced by:: §19.10(i), §19.25(iv), §23.22(ii)
Permalink:: http://dlmf.nist.gov/20.9.i
See also:: Annotations for §20.9 and Ch.20

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 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	$\displaystyle K\left(k\right)$	$\displaystyle=\tfrac{1}{2}\pi{\theta_{3}}^{2}\left(0\middle\|\tau\right),$
20.9.2	$\displaystyle K'\left(k\right)$	$\displaystyle=-i\tau K\left(k\right),$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : 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		$R_{F}\left(\frac{{\theta_{2}}^{2}\left(z,q\right)}{{\theta_{2}}^{2}\left(0,q% \right)},\frac{{\theta_{3}}^{2}\left(z,q\right)}{{\theta_{3}}^{2}\left(0,q% \right)},\frac{{\theta_{4}}^{2}\left(z,q\right)}{{\theta_{4}}^{2}\left(0,q% \right)}\right)=\frac{\theta_{1}'\left(0,q\right)}{\theta_{1}\left(z,q\right)}z,$
ⓘ 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 Ch.20

20.9.4		$R_{F}\left(0,{\theta_{3}}^{4}\left(0,q\right),{\theta_{4}}^{4}\left(0,q\right)% \right)=\tfrac{1}{2}\pi,$
ⓘ 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, $\pi$ : 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		$\exp\left(-\frac{\pi R_{F}\left(0,k^{2},1\right)}{R_{F}\left(0,{k^{\prime}}^{2% },1\right)}\right)=q.$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : 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

§20.9(ii) Elliptic Functions and Modular Functions

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Keywords:: Jacobian elliptic functions, Jacobi’s inversion formula, Jacobi’s inversion problem for elliptic functions, Weierstrass elliptic functions, elliptic modular function, modular functions, relations to other functions, relations to theta functions, theta functions
Notes:: See Walker (1996, p. 156), Whittaker and Watson (1927, p. 481), Serre (1973, p. 109), and McKean and Moll (1999, §3.9).
Referenced by:: §20.11(iii)
Permalink:: http://dlmf.nist.gov/20.9.ii
See also:: Annotations for §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).

§20.9(iii) Riemann Zeta Function

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Keywords:: Riemann zeta function, relations to other functions, theta functions
Permalink:: http://dlmf.nist.gov/20.9.iii
See also:: Annotations for §20.9 and Ch.20

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

20.8 Watson’s Expansions 20.10 Integrals

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§20.9 Relations to Other Functions

Contents

§20.9(i) Elliptic Integrals

§20.9(ii) Elliptic Functions and Modular Functions

§20.9(iii) Riemann Zeta Function