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

§20.4 Values at = 0

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

§20.4(i) Functions and First Derivatives
§20.4(ii) Higher Derivatives

§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 $\theta_{1}\left(0,q\right)=\theta_{2}'\left(0,q\right)=\theta_{3}'\left(0,q% \right)=\theta_{4}'\left(0,q\right)=0,$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : 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 $\theta_{1}'\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)^{3}=2q^{1/4}{\left(q^{2};q^{2}\right)_{\infty}}^{3},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : -Pochhammer symbol (or -shifted factorial), : integer and : 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 $\theta_{2}\left(0,q\right)=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}% \right)\left(1+q^{2n}\right)^{2},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\theta_{3}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1+q^{2n-1}\right)^{2},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\theta_{4}\left(0,q\right)=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)% \left(1-q^{2n-1}\right)^{2}.$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\theta_{1}'\left(0,q\right)=\theta_{2}\left(0,q\right)\theta_{3}\left(0,q% \right)\theta_{4}\left(0,q\right).$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : 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
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 $\theta_{1}''\left(0,q\right)=\theta_{2}'''\left(0,q\right)=\theta_{3}'''\left(% 0,q\right)=\theta_{4}'''\left(0,q\right)=0.$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : 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 $\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=-1+24\sum_{n% =1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}.$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\frac{\theta_{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}=-1-8\sum_{n=1}% ^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\frac{\theta_{3}''\left(0,q\right)}{\theta_{3}\left(0,q\right)}=-8\sum_{n=1}^{% \infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}},$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\frac{\theta_{4}''\left(0,q\right)}{\theta_{4}\left(0,q\right)}=8\sum_{n=1}^{% \infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}.$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, : integer and : 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 $\frac{\theta_{1}'''\left(0,q\right)}{\theta_{1}'\left(0,q\right)}=\frac{\theta% _{2}''\left(0,q\right)}{\theta_{2}\left(0,q\right)}+\frac{\theta_{3}''\left(0,% q\right)}{\theta_{3}\left(0,q\right)}+\frac{\theta_{4}''\left(0,q\right)}{% \theta_{4}\left(0,q\right)}.$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : 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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§20.4 Values at = 0

Contents

§20.4(i) Functions and First Derivatives

Jacobi’s Identity

§20.4(ii) Higher Derivatives