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

§23.18 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).
Keywords:: modular functions
Referenced by:: §20.14
Permalink:: http://dlmf.nist.gov/23.18

¶ Elliptic Modular Function

$\mathop{\lambda\/}\nolimits\!\left(\mathcal{A}\tau\right)$ equals

23.18.1

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

$1-\mathop{\lambda\/}\nolimits\!\left(\tau\right),$

$\frac{1}{\mathop{\lambda\/}\nolimits\!\left(\tau\right)},$

$\frac{1}{1-\mathop{\lambda\/}\nolimits\!\left(\tau\right)},$

$\frac{\mathop{\lambda\/}\nolimits\!\left(\tau\right)}{\mathop{\lambda\/}% \nolimits\!\left(\tau\right)-1},$

$1-\frac{1}{\mathop{\lambda\/}\nolimits\!\left(\tau\right)},$

Symbols:: $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : 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

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

Here e and o are generic symbols for even and odd integers, respectively. In particular, if , and are all even, then

23.18.3 $\mathop{\lambda\/}\nolimits\!\left(\mathcal{A}\tau\right)=\mathop{\lambda\/}% \nolimits\!\left(\tau\right),$

Symbols:: $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : elliptic modular function, $\tau$ : complex variable and $\mathcal{A}$ : bilinear transformation
Permalink:: http://dlmf.nist.gov/23.18.E3
Encodings:: TeX, pMML, png

and $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ is a cusp form of level zero for the corresponding subgroup of SL $(2,\Integer)$ .

¶ Klein’s Complete Invariant

23.18.4 $\mathop{J\/}\nolimits\!\left(\mathcal{A}\tau\right)=\mathop{J\/}\nolimits\!% \left(\tau\right).$

Symbols:: $\mathop{J\/}\nolimits\!\left(\tau\right)$ : 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

$\mathop{J\/}\nolimits\!\left(\tau\right)$ is a modular form of level zero for SL $(2,\Integer)$ .

¶ Dedekind’s Eta Function

23.18.5 $\mathop{\eta\/}\nolimits\!\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A}% )\left(-i(c\tau+d)\right)^{{1/2}}\mathop{\eta\/}\nolimits\!\left(\tau\right),$

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), $\tau$ : 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

where the square root has its principal value and

23.18.6 $\varepsilon(\mathcal{A})=\mathop{\exp\/}\nolimits\!\left(\pi i\left(\frac{a+d}% {12c}+s(-d,c)\right)\right),$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathcal{A}$ : bilinear transformation, : integer, : integer, : integer and $\varepsilon(\mathcal{A})$ : function
Permalink:: http://dlmf.nist.gov/23.18.E6
Encodings:: TeX, pMML, png

23.18.7 ${s(d,c)=\sum_{{\substack{r=1\\ (r,c)=1}}}^{{c-1}}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}\right% \rfloor-\frac{1}{2}\right),}$

Symbols:: $\left\lfloor x\right\rfloor$ : floor of , : open interval, : integer and : integer
Referenced by:: §23.18
Permalink:: http://dlmf.nist.gov/23.18.E7
Encodings:: TeX, pMML, png

Here the notation means that the sum is confined to those values of that are relatively prime to . See §27.14(iii) and Apostol (1990, pp. 48 and 51–53). Note that $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ is of level $\tfrac{1}{2}$ .

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