§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 $\mathop{\wp\/}\nolimits\!\left(z\right)-e_{j}=\left(\frac{v{\mathop{\theta _{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)\mathop{\theta _{{j+1}}\/}\nolimits\!\left(v,q\right)}{z\mathop{\theta _{{1}}\/}\nolimits\!\left(v,q\right)\mathop{\theta _{{j+1}}\/}\nolimits\!\left(0,q\right)}\right)^{2}$ , $j=1,2,3$ , 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 $z$ ’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 $\mathop{K\/}\nolimits\!\left(k\right)=\mathop{R_{F}\/}\nolimits\!\left(0,\frac{{\mathop{\theta _{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)}{{\mathop{\theta _{{3}}\/}\nolimits^{{4}}}\!\left(0,q\right)},1\right)={\mathop{\theta _{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)\mathop{R_{F}\/}\nolimits\!\left(0,{\mathop{\theta _{{3}}\/}\nolimits^{{4}}}\!\left(0,q\right),{\mathop{\theta _{{4}}\/}\nolimits^{{4}}}\!\left(0,q\right)\right)$ , 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 $q$ by Erdélyi et al. (1953b, 13.19(4)).
Keywords:: elliptic integrals, theta functions
Referenced by:: §19.10(i), §19.25(iv), ¶ ‣ §23.22(ii)
Permalink:: http://dlmf.nist.gov/20.9.i

With $k$ 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 $k$ : modulus
Referenced by:: §20.11(iii), §20.9(ii), §23.6(ii)
Permalink:: http://dlmf.nist.gov/20.9.E1
Encodings:: TeX, pMML, png

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 $k$ : 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
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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, $z$ : complex and $q$ : nome
Referenced by:: §19.25(v), §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E3
Encodings:: TeX, pMML, png

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 $q$ : nome
Referenced by:: §20.9(i)
Permalink:: http://dlmf.nist.gov/20.9.E4
Encodings:: TeX, pMML, png

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, $q$ : nome and $k$ : 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 $k$ 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

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.

§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