# §23.17 Elementary Properties

## §23.17(i) Special Values

 23.17.1 $\displaystyle\mathop{\lambda\/}\nolimits\!\left(i\right)$ $\displaystyle=\tfrac{1}{2},$ $\displaystyle\mathop{\lambda\/}\nolimits\!\left(\textstyle e^{\pi i/3}\right)$ $\displaystyle=e^{\pi i/3},$
 23.17.2 $\displaystyle\mathop{J\/}\nolimits\!\left(i\right)$ $\displaystyle=1,$ $\displaystyle\mathop{J\/}\nolimits\!\left(\textstyle e^{\pi i/3}\right)$ $\displaystyle=0,$
 23.17.3 $\displaystyle\mathop{\eta\/}\nolimits\!\left(i\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{4}\right)}{2% \pi^{3/4}},$ $\displaystyle\mathop{\eta\/}\nolimits\!\left(\textstyle e^{\pi i/3}\right)$ $\displaystyle=\frac{3^{1/8}\left(\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3% }\right)\right)^{3/2}}{2\pi}e^{\pi i/24}.$

For further results for $\mathop{J\/}\nolimits\!\left(\tau\right)$ see Cohen (1993, p. 376).

## §23.17(ii) Power and Laurent Series

When $|q|<1$

 23.17.4 $\mathop{\lambda\/}\nolimits\!\left(\tau\right)=16q(1-8q+44q^{2}+\cdots),$ Symbols: $\mathop{\lambda\/}\nolimits\!\left(\NVar{\tau}\right)$: elliptic modular function, $q$: nome and $\tau$: complex variable Permalink: http://dlmf.nist.gov/23.17.E4 Encodings: TeX, pMML, png See also: Annotations for 23.17(ii)
 23.17.5 $1728\!\mathop{J\/}\nolimits\!\left(\tau\right)=q^{-2}+744+1\;96884q^{2}+214\;9% 3760q^{4}+\cdots,$ Symbols: $\mathop{J\/}\nolimits\!\left(\NVar{\tau}\right)$: Klein’s complete invariant, $q$: nome and $\tau$: complex variable Referenced by: §23.17(ii) Permalink: http://dlmf.nist.gov/23.17.E5 Encodings: TeX, pMML, png See also: Annotations for 23.17(ii)
 23.17.6 $\mathop{\eta\/}\nolimits\!\left(\tau\right)=\sum_{n=-\infty}^{\infty}(-1)^{n}q% ^{(6n+1)^{2}/12}.$

In (23.17.5) for terms up to $q^{48}$ see Zuckerman (1939), and for terms up to $q^{100}$ see van Wijngaarden (1953). See also Apostol (1990, p. 22).

## §23.17(iii) Infinite Products

 23.17.7 $\mathop{\lambda\/}\nolimits\!\left(\tau\right)=16q\prod_{n=1}^{\infty}\left(% \frac{1+q^{2n}}{1+q^{2n-1}}\right)^{8},$
 23.17.8 $\mathop{\eta\/}\nolimits\!\left(\tau\right)=q^{1/12}\prod_{n=1}^{\infty}(1-q^{% 2n}),$ Symbols: $\mathop{\eta\/}\nolimits\!\left(\NVar{\tau}\right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable Referenced by: §23.15(ii), §23.19 Permalink: http://dlmf.nist.gov/23.17.E8 Encodings: TeX, pMML, png See also: Annotations for 23.17(iii)

with $q^{1/12}=e^{i\pi\tau/12}$.