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

$\lambda\left(\tau\right),$

$1-\lambda\left(\tau\right),$

$\frac{1}{\lambda\left(\tau\right)},$

$\frac{1}{1-\lambda\left(\tau\right)},$

$\frac{\lambda\left(\tau\right)}{\lambda\left(\tau\right)-1},$

$1-\frac{1}{\lambda\left(\tau\right)},$

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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

$\begin{bmatrix}\mathrm{o}&\mathrm{e}\\ \mathrm{e}&\mathrm{o}\end{bmatrix},$

$\begin{bmatrix}\mathrm{e}&\mathrm{o}\\ \mathrm{o}&\mathrm{e}\end{bmatrix},$

$\begin{bmatrix}\mathrm{o}&\mathrm{e}\\ \mathrm{o}&\mathrm{o}\end{bmatrix},$

$\begin{bmatrix}\mathrm{e}&\mathrm{o}\\ \mathrm{o}&\mathrm{o}\end{bmatrix},$

$\begin{bmatrix}\mathrm{o}&\mathrm{o}\\ \mathrm{e}&\mathrm{o}\end{bmatrix},$

$\begin{bmatrix}\mathrm{o}&\mathrm{o}\\ \mathrm{o}&\mathrm{e}\end{bmatrix}.$

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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 , and are all even, then

23.18.3 $\lambda\left(\mathcal{A}\tau\right)=\lambda\left(\tau\right),$

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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 $J\left(\mathcal{A}\tau\right)=J\left(\tau\right).$

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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 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$

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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, : integer, : 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 $\varepsilon(\mathcal{A})=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)% \right),$

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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, : integer, : integer, : 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 ${s(d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}% \right\rfloor-\frac{1}{2}\right),}$

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Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), : integer and : 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 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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