# §23.15 Definitions

## §23.15(i) General Modular Functions

In §§23.1523.19, $k$ 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 $q=\mathop{\exp\/}\nolimits\!\left(-\pi\frac{\mathop{{K^{\prime}}\/}\nolimits\!% \left(k\right)}{\mathop{K\/}\nolimits\!\left(k\right)}\right),$
 23.15.2 $\displaystyle k$ $\displaystyle=\frac{{\mathop{\theta_{2}\/}\nolimits^{2}}\!\left(0,q\right)}{{% \mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q\right)},$ $\displaystyle k^{\prime}$ $\displaystyle=\frac{{\mathop{\theta_{4}\/}\nolimits^{2}}\!\left(0,q\right)}{{% \mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q\right)}.$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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)

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)

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)

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 $f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),$ $\Im{\tau}>0$,

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 $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$, $\mathop{J\/}\nolimits\!\left(\tau\right)$, $\mathop{\eta\/}\nolimits\!\left(\tau\right)$

### Elliptic Modular Function

 23.15.6 $\mathop{\lambda\/}\nolimits\!\left(\tau\right)=\frac{{\mathop{\theta_{2}\/}% \nolimits^{4}}\!\left(0,q\right)}{{\mathop{\theta_{3}\/}\nolimits^{4}}\!\left(% 0,q\right)};$ Defines: $\mathop{\lambda\/}\nolimits\!\left(\NVar{\tau}\right)$: elliptic modular function Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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)

compare also (23.15.2).

### Klein’s Complete Invariant

 23.15.7 $\mathop{J\/}\nolimits\!\left(\tau\right)=\frac{\left({\mathop{\theta_{2}\/}% \nolimits^{8}}\!\left(0,q\right)+{\mathop{\theta_{3}\/}\nolimits^{8}}\!\left(0% ,q\right)+{\mathop{\theta_{4}\/}\nolimits^{8}}\!\left(0,q\right)\right)^{3}}{5% 4\left(\mathop{\theta_{1}\/}\nolimits'\!\left(0,q\right)\right)^{8}},$ Defines: $\mathop{J\/}\nolimits\!\left(\NVar{\tau}\right)$: Klein’s complete invariant Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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)

where (as in §20.2(i))

 23.15.8 $\mathop{\theta_{1}\/}\nolimits'\!\left(0,q\right)=\ifrac{\partial\mathop{% \theta_{1}\/}\nolimits\!\left(z,q\right)}{\partial z}|_{z=0}.$

### Dedekind’s Eta Function (or Dedekind Modular Function)

 23.15.9 $\mathop{\eta\/}\nolimits\!\left(\tau\right)=\left(\tfrac{1}{2}\mathop{\theta_{% 1}\/}\nolimits'\!\left(0,q\right)\right)^{1/3}=e^{i\pi\tau/12}\mathop{\theta_{% 3}\/}\nolimits\!\left(\tfrac{1}{2}\pi(1+\tau)\middle|3\tau\right).$

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.