§23.17 Elementary Properties

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§23.17(i) Special Values

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23.17.1	$\displaystyle\lambda\left(i\right)$	$\displaystyle=\tfrac{1}{2},$
23.17.1	$\displaystyle\lambda\left(\textstyle e^{\pi i/3}\right)$	$\displaystyle=e^{\pi i/3},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function Permalink: http://dlmf.nist.gov/23.17.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.17(i), §23.17 and Ch.23

23.17.2	$\displaystyle J\left(i\right)$	$\displaystyle=1,$
23.17.2	$\displaystyle J\left(\textstyle e^{\pi i/3}\right)$	$\displaystyle=0,$
ⓘ Symbols: $J\left(\NVar{\tau}\right)$ : Klein’s complete invariant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/23.17.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.17(i), §23.17 and Ch.23

23.17.3	$\displaystyle\eta\left(i\right)$	$\displaystyle=\frac{\Gamma\left(\tfrac{1}{4}\right)}{2\pi^{3/4}},$
23.17.3	$\displaystyle\eta\left(\textstyle e^{\pi i/3}\right)$	$\displaystyle=\frac{3^{1/8}\left(\Gamma\left(\tfrac{1}{3}\right)\right)^{3/2}}% {2\pi}e^{\pi i/24}.$
ⓘ Symbols: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Referenced by: §22.20(iv) Permalink: http://dlmf.nist.gov/23.17.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §23.17(i), §23.17 and Ch.23

For further results for $J\left(\tau\right)$ see Cohen (1993, p. 376).

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Keywords:: Laurent series, modular functions, power series
Notes:: Combine §23.15(ii) with the $q$ -expansions of the theta functions obtained by setting $z=0$ in §20.2(i).
Permalink:: http://dlmf.nist.gov/23.17.ii
See also:: Annotations for §23.17 and Ch.23

When $|q|<1$

23.17.4		$\lambda\left(\tau\right)=16q(1-8q+44q^{2}+\cdots),$
ⓘ Symbols: $\lambda\left(\NVar{\tau}\right)$ : 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 and Ch.23

23.17.5		$1728J\left(\tau\right)=q^{-2}+744+1\;96884q^{2}+214\;93760q^{4}+\cdots,$
ⓘ Symbols: $J\left(\NVar{\tau}\right)$ : 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 and Ch.23

23.17.6		$\eta\left(\tau\right)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(6n+1)^{2}/12}.$
ⓘ Symbols: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $q$ : nome, $n$ : integer and $\tau$ : complex variable Permalink: http://dlmf.nist.gov/23.17.E6 Encodings: TeX, pMML, png See also: Annotations for §23.17(ii), §23.17 and Ch.23

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).

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23.17.7		$\lambda\left(\tau\right)=16q\prod_{n=1}^{\infty}\left(\frac{1+q^{2n}}{1+q^{2n-% 1}}\right)^{8},$
ⓘ Symbols: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function, $q$ : nome, $n$ : integer and $\tau$ : complex variable Permalink: http://dlmf.nist.gov/23.17.E7 Encodings: TeX, pMML, png See also: Annotations for §23.17(iii), §23.17 and Ch.23

23.17.8		$\eta\left(\tau\right)=q^{1/12}\prod_{n=1}^{\infty}(1-q^{2n}),$
ⓘ Symbols: $\eta\left(\NVar{\tau}\right)$ : 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), §23.17 and Ch.23

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