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

§23.15 Definitions

Keywords:: modular functions, theta functions
Referenced by:: §20.7(viii), §23.15(i), §27.14(iv)
Permalink:: http://dlmf.nist.gov/23.15

§23.15(i) General Modular Functions
§23.15(ii) Functions $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ , $\mathop{J\/}\nolimits\!\left(\tau\right)$ , $\mathop{\eta\/}\nolimits\!\left(\tau\right)$

§23.15(i) General Modular Functions

Notes:: See Apostol (1990, Chapters 1, 2), Walker (1996, Chapter 7), McKean and Moll (1999, Chapters 4, 6).
Keywords:: SL $(2,\Integer)$ bilinear transformation, bilinear transformation, modular functions
Permalink:: http://dlmf.nist.gov/23.15.i

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

23.15.1 $q=\mathop{\exp\/}\nolimits\!\left(-\pi\frac{\mathop{{K^{{\prime}}}\/}\nolimits% \!\left(k\right)}{\mathop{K\/}\nolimits\!\left(k\right)}\right),$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\exp\/}\nolimits z$ : exponential function, : nome and : modulus
Permalink:: http://dlmf.nist.gov/23.15.E1
Encodings:: TeX, pMML, png

23.15.2

$k=\frac{{\mathop{\theta_{{2}}\/}\nolimits^{{2}}}\!\left(0,q\right)}{{\mathop{% \theta_{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)},$

$k^{{\prime}}=\frac{{\mathop{\theta_{{4}}\/}\nolimits^{{2}}}\!\left(0,q\right)}% {{\mathop{\theta_{{3}}\/}\nolimits^{{2}}}\!\left(0,q\right)}.$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, : nome, : 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

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, : integer, : integer, : integer and : integer
Referenced by:: §20.11(i), ¶ ‣ §23.18
Permalink:: http://dlmf.nist.gov/23.15.E3
Encodings:: TeX, pMML, png

in which are integers, with

23.15.4

Symbols:: : integer, : integer, : integer and : integer
Referenced by:: §20.11(i)
Permalink:: http://dlmf.nist.gov/23.15.E4
Encodings:: TeX, pMML, png

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

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

23.15.5 $f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),$ $\imagpart{\tau}>0$ ,

Symbols:: $\imagpart{}$ : imaginary part, $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, : integer, : integer, $f(\tau)$ : modular function, $c_{{\mathcal{A}}}$ : constant and $\ell$ : level
Permalink:: http://dlmf.nist.gov/23.15.E5
Encodings:: TeX, pMML, png

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

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

¶ 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:: $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : elliptic modular function
Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, : 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

compare also (23.15.2).

¶ Klein’s Complete Invariant

23.15.7 $\mathop{J\/}\nolimits\!\left(\tau\right)=\frac{\left({\mathop{\theta_{{2}}\/}% \nolimits^{{8}}}\!\left(0,q\right)+{\mathop{\theta_{{3}}\/}\nolimits^{{8}}}\!% \left(0,q\right)+{\mathop{\theta_{{4}}\/}\nolimits^{{8}}}\!\left(0,q\right)% \right)^{3}}{54\left({\mathop{\theta_{{1}}\/}\nolimits^{{\prime}}}\!\left(0,q% \right)\right)^{8}},$

Defines:: $\mathop{J\/}\nolimits\!\left(\tau\right)$ : Klein’s complete invariant
Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, : nome and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.15.E7
Encodings:: TeX, pMML, png

where (as in §20.2(i))

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : nome and : complex
Permalink:: http://dlmf.nist.gov/23.15.E8
Encodings:: TeX, pMML, png

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

Keywords:: eta function, modular functions, theta functions

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

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : base of exponential function, : 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

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.