23 Weierstrass Elliptic and Modular Functions Modular Functions 23.14 Integrals 23.16 Graphics

§23.15 Definitions

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Keywords:: definitions, modular functions, relations to other functions, relations to theta functions, theta functions
Referenced by:: §20.7(viii), §20.7(viii), §23.15(i), §27.14(iv)
Permalink:: http://dlmf.nist.gov/23.15
See also:: Annotations for Ch.23

§23.15(i) General Modular Functions
§23.15(ii) Functions $\lambda\left(\tau\right)$ , $J\left(\tau\right)$ , $\eta\left(\tau\right)$

§23.15(i) General Modular Functions

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Keywords:: SL $(2,\mathbb{Z})$ , SL $(2,\mathbb{Z})$ bilinear transformation, analytic properties, bilinear transformation, cusp form, general, level, modular form, modular functions
Notes:: See Apostol (1990, Chapters 1, 2), Walker (1996, Chapter 7), McKean and Moll (1999, Chapters 4, 6).
Permalink:: http://dlmf.nist.gov/23.15.i
See also:: Annotations for §23.15 and Ch.23

In §§23.15–23.19, $k$ and $k^{\prime}$ $(\in\mathbb{C})$ denote the Jacobi modulus and complementary modulus, respectively, and $q=e^{i\pi\tau}$ ( $\Im\tau>0$ ) denotes the nome; compare §§20.1 and 22.1. Thus

23.15.1		$q=\exp\left(-\pi\frac{{K^{\prime}}\left(k\right)}{K\left(k\right)}\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\exp\NVar{z}$ : exponential function, $q$ : nome and $k$ : modulus Permalink: http://dlmf.nist.gov/23.15.E1 Encodings: TeX, pMML, png See also: Annotations for §23.15(i), §23.15 and Ch.23

23.15.2	$\displaystyle k$	$\displaystyle=\frac{{\theta_{2}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,% q\right)},$
23.15.2	$\displaystyle k^{\prime}$	$\displaystyle=\frac{{\theta_{4}}^{2}\left(0,q\right)}{{\theta_{3}}^{2}\left(0,% q\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $k$ : modulus and $k^{\prime}$ : complementary modulus Referenced by: §20.9(ii), §23.15(ii) Permalink: http://dlmf.nist.gov/23.15.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.15(i), §23.15 and Ch.23

Also $\mathcal{A}$ denotes a bilinear transformation on $\tau$ , given by

23.15.3		$\mathcal{A}\tau=\frac{a\tau+b}{c\tau+d},$
ⓘ Symbols: $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $a$ : integer, $b$ : integer, $c$ : integer and $d$ : integer Referenced by: §20.11(i), §23.18 Permalink: http://dlmf.nist.gov/23.15.E3 Encodings: TeX, pMML, png See also: Annotations for §23.15(i), §23.15 and Ch.23

in which $a, b, c, d$ are integers, with

23.15.4		$ad-bc=1.$
ⓘ Symbols: $a$ : integer, $b$ : integer, $c$ : integer and $d$ : integer Referenced by: §20.11(i) Permalink: http://dlmf.nist.gov/23.15.E4 Encodings: TeX, pMML, png See also: Annotations for §23.15(i), §23.15 and Ch.23

The set of all bilinear transformations of this form is denoted by SL $(2,\mathbb{Z})$ (Serre (1973, p. 77)).

A modular function $f(\tau)$ is a function of $\tau$ that is meromorphic in the half-plane $\Im\tau>0$ , and has the property that for all $\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})$ , or for all $\mathcal{A}$ belonging to a subgroup of SL $(2,\mathbb{Z})$ ,

23.15.5		$f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),$
$\Im\tau>0$ ,
ⓘ Symbols: $\Im$ : imaginary part, $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $c$ : integer, $d$ : integer, $f(\tau)$ : modular function, $c_{\mathcal{A}}$ : constant and $\ell$ : level Permalink: http://dlmf.nist.gov/23.15.E5 Encodings: TeX, pMML, png See also: Annotations for §23.15(i), §23.15 and Ch.23

where $c_{\mathcal{A}}$ is a constant depending only on $\mathcal{A}$ , and $\ell$ (the level) is an integer or half an odd integer. (Some references refer to $2\ell$ as the level). If, as a function of $q$ , $f(\tau)$ is analytic at $q=0$ , then $f(\tau)$ is called a modular form. If, in addition, $f(\tau)\to 0$ as $q\to 0$ , then $f(\tau)$ is called a cusp form.

§23.15(ii) Functions $\lambda\left(\tau\right)$ , $J\left(\tau\right)$ , $\eta\left(\tau\right)$

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Keywords:: modular functions, notation
Notes:: For (23.15.9) use (20.5.3) and (23.17.8).
Referenced by:: §23.17(ii), §23.22(i)
Permalink:: http://dlmf.nist.gov/23.15.ii
See also:: Annotations for §23.15 and Ch.23

Elliptic Modular Function

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See also:: Annotations for §23.15(ii), §23.15 and Ch.23

23.15.6		$\lambda\left(\tau\right)=\frac{{\theta_{2}}^{4}\left(0,q\right)}{{\theta_{3}}^% {4}\left(0,q\right)};$
ⓘ Defines: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome and $\tau$ : complex variable Referenced by: §20.9(ii), §23.17(iii) Permalink: http://dlmf.nist.gov/23.15.E6 Encodings: TeX, pMML, png See also: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

compare also (23.15.2).

Klein’s Complete Invariant

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Keywords:: Klein’s complete invariant, modular functions
See also:: Annotations for §23.15(ii), §23.15 and Ch.23

23.15.7		$J\left(\tau\right)=\frac{\left({\theta_{2}}^{8}\left(0,q\right)+{\theta_{3}}^{% 8}\left(0,q\right)+{\theta_{4}}^{8}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},$
ⓘ Defines: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome and $\tau$ : complex variable Permalink: http://dlmf.nist.gov/23.15.E7 Encodings: TeX, pMML, png See also: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

where (as in §20.2(i))

23.15.8		$\theta_{1}'\left(0,q\right)=\ifrac{\partial\theta_{1}\left(z,q\right)}{% \partial z}\|_{z=0}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $q$ : nome and $z$ : complex Permalink: http://dlmf.nist.gov/23.15.E8 Encodings: TeX, pMML, png See also: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

Dedekind’s Eta Function (or Dedekind Modular Function)

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Keywords:: Dedekind modular function, Dedekind’s eta function, definitions, eta function, modular functions, relations to other functions, relations to theta functions, theta functions
See also:: Annotations for §23.15(ii), §23.15 and Ch.23

23.15.9		$\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/% 3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle\|3\tau\right).$
ⓘ Symbols: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\theta_{\NVar{j}}\left(\NVar{z}\middle\|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $q$ : nome and $\tau$ : complex variable Referenced by: §23.15(ii), §23.15(ii), §23.17(iii) Permalink: http://dlmf.nist.gov/23.15.E9 Encodings: TeX, pMML, png See also: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when $\tau$ lies on the positive imaginary axis the cube root is real and positive. See also 27.14.12.

23.14 Integrals 23.16 Graphics

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