# §20.4 Values at $z$ = 0

## §20.4(i) Functions and First Derivatives

 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

 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

 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