20 Theta Functions Properties 20.8 Watson’s Expansions 20.10 Integrals

§20.9 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/20.9

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

§20.9(i) Elliptic Integrals

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 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{j}=\left(\frac{v{\mathop{\theta_{{1% }}\/}\nolimits^{{\prime}}}\!\left(0,q\right)\mathop{\theta_{{j+1}}\/}\nolimits% \!\left(v,q\right)}{z\mathop{\theta_{{1}}\/}\nolimits\!\left(v,q\right)\mathop% {\theta_{{j+1}}\/}\nolimits\!\left(0,q\right)}\right)^{2}$ , , where $v=\pi z/(2\omega_{1})$ . Then by application of (19.25.35) and use of the properties that $\mathop{R_{F}\/}\nolimits$ is homogenous and of degree $-\tfrac{1}{2}$ in its three variables (§§19.16(ii), 19.16(iii)), we derive $z=\frac{z\mathop{\theta_{{1}}\/}\nolimits\!\left(v,q\right)}{v{\mathop{\theta_% {{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}\mathop{R_{F}\/}\nolimits\!% \left(\frac{{\mathop{\theta_{{2}}\/}\nolimits^{{2}}}\!\left(v,q\right)}{{% \mathop{\theta_{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right)},\frac{{\mathop{% \theta_{{3}}\/}\nolimits^{{2}}}\!\left(v,q\right)}{{\mathop{\theta_{{3}}\/}% \nolimits^{{2}}}\!\left(0,q\right)},\frac{{\mathop{\theta_{{4}}\/}\nolimits^{{% 2}}}\!\left(v,q\right)}{{\mathop{\theta_{{4}}\/}\nolimits^{{2}}}\!\left(0,q% \right)}\right)$ . This equation becomes (20.9.3) when the ’s are cancelled and is renamed . For (20.9.4), from (19.25.1) and Erdélyi et al. (1953b, 13.20(11)) we have $\mathop{K\/}\nolimits\!\left(k\right)=\mathop{R_{F}\/}\nolimits\!\left(0,\frac% {{\mathop{\theta_{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)}{{\mathop{\theta_{% {3}}\/}\nolimits^{{4}}}\!\left(0,q\right)},1\right)={\mathop{\theta_{{3}}\/}% \nolimits^{{2}}}\!\left(0,q\right)\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{% \theta_{{3}}\/}\nolimits^{{4}}}\!\left(0,q\right),{\mathop{\theta_{{4}}\/}% \nolimits^{{4}}}\!\left(0,q\right)\right)$ , where the second equality uses the homogeneity and symmetry of $\mathop{R_{F}\/}\nolimits$ . Comparison with (20.9.2) proves (20.9.4). For (20.9.5), by (19.25.1) the left side is $\mathop{\exp\/}\nolimits\!\left(-\pi\!\mathop{K\/}\nolimits\!\left(k^{{\prime}% }\right)/\mathop{K\/}\nolimits\!\left(k\right)\right)$ , which equals by Erdélyi et al. (1953b, 13.19(4)).
Keywords:: elliptic integrals, theta functions
Referenced by:: §19.10(i), §19.25(iv), ¶ ‣ §23.22(ii)
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With 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 : modulus
Referenced by:: §20.11(iii), §20.9(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/20.9.E1
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and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions:

20.9.2

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

${\mathop{K\/}\nolimits^{{\prime}}}\!\left(k\right)=-i\tau\mathop{K\/}\nolimits% \!\left(k\right),$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\tau$ : lattice parameter and : 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
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together with (22.2.1).

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

20.9.3 $\mathop{R_{F}\/}\nolimits\!\left(\frac{{\mathop{\theta_{{2}}\/}\nolimits^{{2}}% }\!\left(z,q\right)}{{\mathop{\theta_{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right% )},\frac{{\mathop{\theta_{{3}}\/}\nolimits^{{2}}}\!\left(z,q\right)}{{\mathop{% \theta_{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)},\frac{{\mathop{\theta_{{4}}% \/}\nolimits^{{2}}}\!\left(z,q\right)}{{\mathop{\theta_{{4}}\/}\nolimits^{{2}}% }\!\left(0,q\right)}\right)=\frac{{\mathop{\theta_{{1}}\/}\nolimits^{{\prime}}% }\!\left(0,q\right)}{\mathop{\theta_{{1}}\/}\nolimits\!\left(z,q\right)}z,$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, : complex and : nome
Referenced by:: §19.25(v), §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E3
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20.9.4 $\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\theta_{{3}}\/}\nolimits^{{4}}}\!% \left(0,q\right),{\mathop{\theta_{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)% \right)=\tfrac{1}{2}\pi,$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and : nome
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E4
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20.9.5 $\mathop{\exp\/}\nolimits\!\left(-\frac{\pi\mathop{R_{F}\/}\nolimits\!\left(0,k% ^{2},1\right)}{\mathop{R_{F}\/}\nolimits\!\left(0,{k^{{\prime}}}^{2},1\right)}% \right)=q.$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\exp\/}\nolimits z$ : exponential function, : nome and : modulus
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E5
Encodings:: TeX, pMML, png

§20.9(ii) Elliptic Functions and Modular Functions

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

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 $\tau$ (or ) 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

Keywords:: Riemann zeta function, theta functions
Permalink:: http://dlmf.nist.gov/20.9.iii

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 20.8 Watson’s Expansions 20.10 Integrals

§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