# 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 $a-1,b,c$, and $d-1$ are all even, then

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

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

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

# Dedekind’s Eta Function

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

where the square root has its principal value and

 23.18.6 $\varepsilon(\mathcal{A})=\mathop{\exp\/}\nolimits\!\left(\pi i\left(\frac{a+d}% {12c}+s(-d,c)\right)\right),$
 23.18.7 ${s(d,c)=\sum_{\substack{r=1\\ (r,c)=1}}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\left\lfloor\frac{dr}{c}\right% \rfloor-\frac{1}{2}\right),}$ $c>0$.

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