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DLMF: §20.4 Values at 𝑧 = 0 ‣ Properties ‣ Chapter 20 Theta Functions

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20 Theta Functions Properties 20.3 Graphics 20.5 Infinite Products and Related Results

§20.4 Values at $z$ = 0

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Keywords:: derivatives, theta functions, values at $z=0$
Permalink:: http://dlmf.nist.gov/20.4
See also:: Annotations for Ch.20

Contents

§20.4(i) Functions and First Derivatives

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Notes:: See Walker (1996, pp. 90–92), Whittaker and Watson (1927, pp. 470–471), and Lawden (1989, pp. 12–15). Note that these are special cases of (20.5.1)–(20.5.4).
Permalink:: http://dlmf.nist.gov/20.4.i
See also:: Annotations for §20.4 and Ch.20

20.4.1		$\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)=\theta_{4}'\left(0,q\right)=0,$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome Permalink: http://dlmf.nist.gov/20.4.E1 Encodings: TeX, pMML, png See also: Annotations for §20.4(i), §20.4 and Ch.20

20.4.2		$\displaystyle\theta_{1}'\left(0,q\right)$	$\displaystyle=2q^{1/4}\prod_{n=1}^{\infty}\left(1-q^{2n}\right)^{3}=2q^{1/4}{% \left(q^{2};q^{2}\right)_{\infty}}^{3},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $n$ : integer and $q$ : nome Referenced by: Erratum (V1.0.28) for Equation (20.4.2) Permalink: http://dlmf.nist.gov/20.4.E2 Encodings: TeX, pMML, png Addition (effective with 1.0.28): The representation in terms of ${\left(q^{2};q^{2}\right)_{\infty}}^{3}$ was added to this equation. See also: Annotations for §20.4(i), §20.4 and Ch.20
20.4.3		$\displaystyle\theta_{2}\left(0,q\right)$	$\displaystyle=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+% q^{2n}\right)^{2},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Referenced by: (20.10.3) Permalink: http://dlmf.nist.gov/20.4.E3 Encodings: TeX, pMML, png See also: Annotations for §20.4(i), §20.4 and Ch.20
20.4.4		$\displaystyle\theta_{3}\left(0,q\right)$	$\displaystyle=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}% \right)^{2},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Referenced by: (20.10.2) Permalink: http://dlmf.nist.gov/20.4.E4 Encodings: TeX, pMML, png See also: Annotations for §20.4(i), §20.4 and Ch.20
20.4.5		$\displaystyle\theta_{4}\left(0,q\right)$	$\displaystyle=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1-q^{2n-1}% \right)^{2}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Referenced by: (20.10.1) Permalink: http://dlmf.nist.gov/20.4.E5 Encodings: TeX, pMML, png See also: Annotations for §20.4(i), §20.4 and Ch.20

Jacobi’s Identity

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Keywords:: Jacobi’s identity, Jacobi’s triple product, theta functions
See also:: Annotations for §20.4(i), §20.4 and Ch.20

20.4.6		$\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome A&S Ref: 16.28.6 Referenced by: §20.9(i) Permalink: http://dlmf.nist.gov/20.4.E6 Encodings: TeX, pMML, png See also: Annotations for §20.4(i), §20.4(i), §20.4 and Ch.20

§20.4(ii) Higher Derivatives

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Keywords:: theta functions, values at $z=0$
Notes:: See Whittaker and Watson (1927, p. 471).
Permalink:: http://dlmf.nist.gov/20.4.ii
See also:: Annotations for §20.4 and Ch.20

20.4.7		$\theta_{1}''\left(0,q\right)=\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(% 0,q\right)=\theta_{4}'''\left(0,q\right)=0.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome Permalink: http://dlmf.nist.gov/20.4.E7 Encodings: TeX, pMML, png See also: Annotations for §20.4(ii), §20.4 and Ch.20

20.4.8		$\displaystyle\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}$	$\displaystyle=-1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Referenced by: §23.12 Permalink: http://dlmf.nist.gov/20.4.E8 Encodings: TeX, pMML, png See also: Annotations for §20.4(ii), §20.4 and Ch.20
20.4.9		$\displaystyle\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}$	$\displaystyle=-1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Permalink: http://dlmf.nist.gov/20.4.E9 Encodings: TeX, pMML, png See also: Annotations for §20.4(ii), §20.4 and Ch.20
20.4.10		$\displaystyle\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}\left(0,q\right)}$	$\displaystyle=-8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}},$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Permalink: http://dlmf.nist.gov/20.4.E10 Encodings: TeX, pMML, png See also: Annotations for §20.4(ii), §20.4 and Ch.20
20.4.11		$\displaystyle\frac{\theta_{4}''\left(0,q\right)}{\theta_{4}\left(0,q\right)}$	$\displaystyle=8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $n$ : integer and $q$ : nome Permalink: http://dlmf.nist.gov/20.4.E11 Encodings: TeX, pMML, png See also: Annotations for §20.4(ii), §20.4 and Ch.20

20.4.12		$\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=\frac{\theta% _{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}+\frac{\theta_{3}''\left(0,% q\right)}{\theta_{3}\left(0,q\right)}+\frac{\theta_{4}''\left(0,q\right)}{% \theta_{4}\left(0,q\right)}.$
ⓘ Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $q$ : nome Permalink: http://dlmf.nist.gov/20.4.E12 Encodings: TeX, pMML, png See also: Annotations for §20.4(ii), §20.4 and Ch.20

20.3 Graphics 20.5 Infinite Products and Related Results

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