23 Weierstrass Elliptic and Modular Functions Modular Functions 23.16 Graphics 23.18 Modular Transformations

§23.17 Elementary Properties

Keywords:: modular functions
Permalink:: http://dlmf.nist.gov/23.17

§23.17(i) Special Values
§23.17(ii) Power and Laurent Series
§23.17(iii) Infinite Products

§23.17(i) Special Values

Notes:: See Walker (1996, §7.5).
Keywords:: modular functions
Permalink:: http://dlmf.nist.gov/23.17.i

23.17.1

$\mathop{\lambda\/}\nolimits\!\left(i\right)=\tfrac{1}{2},$

$\mathop{\lambda\/}\nolimits\!\left(\textstyle e^{{\pi i/3}}\right)=e^{{\pi i/3% }},$

Symbols:: : base of exponential function and $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : elliptic modular function
Permalink:: http://dlmf.nist.gov/23.17.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

23.17.2

$\mathop{J\/}\nolimits\!\left(i\right)=1,$

$\mathop{J\/}\nolimits\!\left(\textstyle e^{{\pi i/3}}\right)=0,$

Symbols:: $\mathop{J\/}\nolimits\!\left(\tau\right)$ : Klein’s complete invariant and : base of exponential function
Permalink:: http://dlmf.nist.gov/23.17.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

23.17.3

$\mathop{\eta\/}\nolimits\!\left(i\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{1}{4}\right)}{2\pi^{{3/4}}},$

$\mathop{\eta\/}\nolimits\!\left(\textstyle e^{{\pi i/3}}\right)=\frac{3^{{1/8}% }\left(\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right)\right)^{{3/2}}}{2% \pi}e^{{\pi i/24}}.$

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and : base of exponential function
Referenced by:: ¶ ‣ §22.20(iv)
Permalink:: http://dlmf.nist.gov/23.17.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

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

§23.17(ii) Power and Laurent Series

Notes:: Combine §23.15(ii) with the -expansions of the theta functions obtained by setting in §20.2(i).
Keywords:: Laurent series, modular functions, power series
Permalink:: http://dlmf.nist.gov/23.17.ii

When

23.17.4 $\mathop{\lambda\/}\nolimits\!\left(\tau\right)=16q(1-8q+44q^{2}+\cdots),$

Symbols:: $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$ : elliptic modular function, : nome and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.17.E4
Encodings:: TeX, pMML, png

23.17.5 $1728\!\mathop{J\/}\nolimits\!\left(\tau\right)=q^{{-2}}+744+1\;96884q^{2}+214% \;93760q^{4}+\cdots,$

Symbols:: $\mathop{J\/}\nolimits\!\left(\tau\right)$ : Klein’s complete invariant, : nome and $\tau$ : complex variable
Referenced by:: §23.17(ii)
Permalink:: http://dlmf.nist.gov/23.17.E5
Encodings:: TeX, pMML, png

23.17.6 $\mathop{\eta\/}\nolimits\!\left(\tau\right)=\sum_{{n=-\infty}}^{\infty}(-1)^{n% }q^{{(6n+1)^{2}/12}}.$

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), : nome, : integer and $\tau$ : complex variable
Permalink:: http://dlmf.nist.gov/23.17.E6
Encodings:: TeX, pMML, png

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).