23 Weierstrass Elliptic and Modular Functions Modular Functions 23.17 Elementary Properties 23.19 Interrelations

§23.18 Modular Transformations

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Keywords:: modular functions, modular transformations
Notes:: See Walker (1996, Chapter 7), Ahlfors (1966, pp. 271–274), and Serre (1973, Chapter 7). For (23.18.4)–(23.18.7) see Apostol (1990, pp. 17, 52).
Referenced by:: §20.14
Permalink:: http://dlmf.nist.gov/23.18
See also:: Annotations for Ch.23

Elliptic Modular Function

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See also:: Annotations for §23.18 and Ch.23

$\lambda\left(\mathcal{A}\tau\right)$ equals

23.18.1		$\lambda\left(\tau\right),$
		$1-\lambda\left(\tau\right),$
		$\frac{1}{\lambda\left(\tau\right)},$
		$\frac{1}{1-\lambda\left(\tau\right)},$
		$\frac{\lambda\left(\tau\right)}{\lambda\left(\tau\right)-1},$
		$1-\frac{1}{\lambda\left(\tau\right)},$
ⓘ Symbols: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function and $\tau$ : complex variable Permalink: http://dlmf.nist.gov/23.18.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §23.18, §23.18 and Ch.23

according as the elements $\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ of $\mathcal{A}$ in (23.15.3) have the respective forms

23.18.2		$\begin{bmatrix}\mathrm{o}&\mathrm{e}\\ \mathrm{e}&\mathrm{o}\end{bmatrix},$
		$\begin{bmatrix}\mathrm{e}&\mathrm{o}\\ \mathrm{o}&\mathrm{e}\end{bmatrix},$
		$\begin{bmatrix}\mathrm{o}&\mathrm{e}\\ \mathrm{o}&\mathrm{o}\end{bmatrix},$
		$\begin{bmatrix}\mathrm{e}&\mathrm{o}\\ \mathrm{o}&\mathrm{o}\end{bmatrix},$
		$\begin{bmatrix}\mathrm{o}&\mathrm{o}\\ \mathrm{e}&\mathrm{o}\end{bmatrix},$
		$\begin{bmatrix}\mathrm{o}&\mathrm{o}\\ \mathrm{o}&\mathrm{e}\end{bmatrix}.$
ⓘ Symbols: e: even integers and o: odd integers Permalink: http://dlmf.nist.gov/23.18.E2 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §23.18, §23.18 and Ch.23

Here e and o are generic symbols for even and odd integers, respectively. In particular, if $a-1,b,c$ , and $d-1$ are all even, then

23.18.3		$\lambda\left(\mathcal{A}\tau\right)=\lambda\left(\tau\right),$
ⓘ Symbols: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function, $\tau$ : complex variable and $\mathcal{A}$ : bilinear transformation Permalink: http://dlmf.nist.gov/23.18.E3 Encodings: TeX, pMML, png See also: Annotations for §23.18, §23.18 and Ch.23

and $\lambda\left(\tau\right)$ is a cusp form of level zero for the corresponding subgroup of SL $(2,\mathbb{Z})$ .

Klein’s Complete Invariant

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See also:: Annotations for §23.18 and Ch.23

23.18.4		$J\left(\mathcal{A}\tau\right)=J\left(\tau\right).$
ⓘ Symbols: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant, $\tau$ : complex variable and $\mathcal{A}$ : bilinear transformation Referenced by: §23.18 Permalink: http://dlmf.nist.gov/23.18.E4 Encodings: TeX, pMML, png See also: Annotations for §23.18, §23.18 and Ch.23

$J\left(\tau\right)$ is a modular form of level zero for SL $(2,\mathbb{Z})$ .

Dedekind’s Eta Function

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See also:: Annotations for §23.18 and Ch.23

23.18.5		$\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
ⓘ Symbols: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$ : imaginary unit, $\tau$ : complex variable, $\mathcal{A}$ : bilinear transformation, $c$ : integer, $d$ : integer and $\varepsilon(\mathcal{A})$ : function Permalink: http://dlmf.nist.gov/23.18.E5 Encodings: TeX, pMML, png See also: Annotations for §23.18, §23.18 and Ch.23

where the square root has its principal value and

23.18.6		$\varepsilon(\mathcal{A})=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)% \right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\mathcal{A}$ : bilinear transformation, $a$ : integer, $c$ : integer, $d$ : integer and $\varepsilon(\mathcal{A})$ : function Permalink: http://dlmf.nist.gov/23.18.E6 Encodings: TeX, pMML, png See also: Annotations for §23.18, §23.18 and Ch.23

23.18.7		${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$
$c>0$ .
ⓘ Symbols: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $c$ : integer and $d$ : integer Referenced by: §23.18, Erratum (V1.0.11) for Equation (23.18.7) Permalink: http://dlmf.nist.gov/23.18.E7 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally the sum $\sum_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$ . This additional condition was incorrect and has been removed. Reported 2015-10-05 by Howard Cohl and Tanay Wakhare See also: Annotations for §23.18, §23.18 and Ch.23

Here $s(d,c)$ is a Dedekind sum. See (27.14.11), §27.14(iii), §27.14(iv) and Apostol (1990, pp. 48 and 51–53). Note that $\eta\left(\tau\right)$ is of level $\tfrac{1}{2}$ .

23.17 Elementary Properties 23.19 Interrelations

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