About the Project

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

§23.17 Elementary Properties

ⓘ

Keywords:: elementary properties, modular functions
Permalink:: http://dlmf.nist.gov/23.17
See also:: Annotations for Ch.23

Contents

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

§23.17(i) Special Values

ⓘ

Keywords:: modular functions, special values
Notes:: See Walker (1996, §7.5).
Permalink:: http://dlmf.nist.gov/23.17.i
See also:: Annotations for §23.17 and Ch.23

23.17.1

$\lambda\left(i\right)=\tfrac{1}{2},$

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

ⓘ

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

$J\left(i\right)=1,$

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

ⓘ

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

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

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

ⓘ

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).

§23.17(ii) Power and Laurent Series

ⓘ

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

When

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

ⓘ

Symbols:: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function, : 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, : 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), : nome, : 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).

§23.17(iii) Infinite Products

ⓘ

Keywords:: infinite products, modular functions
Notes:: Combine (23.15.6), (23.15.9), and (20.5.1)–(20.5.3).
Permalink:: http://dlmf.nist.gov/23.17.iii
See also:: Annotations for §23.17 and Ch.23

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, : nome, : 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), : nome, : 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 $q^{1/12}=e^{i\pi\tau/12}$ .

23.16 Graphics 23.18 Modular Transformations

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.7; Release date 2026-06-15 Math-Image version (See MathML) .

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov