23 Weierstrass Elliptic and Modular FunctionsModular Functions23.16 Graphics23.18 Modular Transformations

§23.17 Elementary Properties

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Keywords:

elementary properties, modular functions

Permalink:

http://dlmf.nist.gov/23.17

See also:

Annotations for Ch.23

Contents

1. §23.17(i) Special Values
2. §23.17(ii) Power and Laurent Series
3. §23.17(iii) Infinite Products

§23.17(i) Special Values

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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		$\lambda \left(\mathrm{i}\right)$	$=\frac{1}{2},$
		$\lambda \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)$	$={\mathrm{e}}^{\pi \mathrm{i}/3},$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\lambda \left(\tau \right)$: 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		$J\left(\mathrm{i}\right)$	$=1,$
		$J\left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)$	$=0,$
ⓘ Symbols: $J\left(\tau \right)$: 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		$\eta \left(\mathrm{i}\right)$	$={\displaystyle \frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)}{2{\pi }^{3/4}}},$
		$\eta \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)$	$={\displaystyle \frac{3^{1/8}{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^{3/2}}{2\pi }}{\mathrm{e}}^{\pi \mathrm{i}/24}.$
ⓘ Symbols: $\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{\Gamma }\left(z\right)$: 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

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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

23.17.4		$$\lambda \left(\tau \right)=16q(1-8q+44q^2+\mathrm{\cdots }),$$
ⓘ Symbols: $\lambda \left(\tau \right)$: 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		$$1728J\left(\tau \right)=q^{-2}+744+\mathrm{1\hspace{0.33em}96884}{q}^2+\mathrm{214\hspace{0.33em}93760}{q}^4+\mathrm{\cdots },$$
ⓘ Symbols: $J\left(\tau \right)$: 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		$$\eta \left(\tau \right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{{(6n+1)}^2/12}.$$
ⓘ Symbols: $\eta \left(\tau \right)$: 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).

§23.17(iii) Infinite Products

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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		$$\lambda \left(\tau \right)=16q\prod _{n=1}^{\mathrm{\infty }}{\left(\frac{1+q^{2n}}{1+q^{2n-1}}\right)}^8,$$
ⓘ Symbols: $\lambda \left(\tau \right)$: 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		$$\eta \left(\tau \right)=q^{1/12}\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n}),$$
ⓘ Symbols: $\eta \left(\tau \right)$: 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 $q^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.

23.16 Graphics23.18 Modular Transformations

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