# §20.4 Values at $z$ = 0

## §20.4(i) Functions and First Derivatives

 20.4.1 $\mathop{\theta_{1}\/}\nolimits\!\left(0,q\right)=\mathop{\theta_{2}\/}% \nolimits'\!\left(0,q\right)=\mathop{\theta_{3}\/}\nolimits'\!\left(0,q\right)% =\mathop{\theta_{4}\/}\nolimits'\!\left(0,q\right)=0,$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E1 Encodings: TeX, pMML, png See also: Annotations for 20.4(i)
 20.4.2 $\displaystyle\mathop{\theta_{1}\/}\nolimits'\!\left(0,q\right)$ $\displaystyle=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)^{3},$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $n$: integer and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E2 Encodings: TeX, pMML, png See also: Annotations for 20.4(i) 20.4.3 $\displaystyle\mathop{\theta_{2}\/}\nolimits\!\left(0,q\right)$ $\displaystyle=2q^{1/4}\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+% q^{2n}\right)^{2},$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $n$: integer and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E3 Encodings: TeX, pMML, png See also: Annotations for 20.4(i) 20.4.4 $\displaystyle\mathop{\theta_{3}\/}\nolimits\!\left(0,q\right)$ $\displaystyle=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1+q^{2n-1}% \right)^{2},$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $n$: integer and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E4 Encodings: TeX, pMML, png See also: Annotations for 20.4(i) 20.4.5 $\displaystyle\mathop{\theta_{4}\/}\nolimits\!\left(0,q\right)$ $\displaystyle=\prod\limits_{n=1}^{\infty}\left(1-q^{2n}\right)\left(1-q^{2n-1}% \right)^{2}.$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $n$: integer and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E5 Encodings: TeX, pMML, png See also: Annotations for 20.4(i)

### Jacobi’s Identity

 20.4.6 $\mathop{\theta_{1}\/}\nolimits'\!\left(0,q\right)=\mathop{\theta_{2}\/}% \nolimits\!\left(0,q\right)\mathop{\theta_{3}\/}\nolimits\!\left(0,q\right)% \mathop{\theta_{4}\/}\nolimits\!\left(0,q\right).$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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(ii) Higher Derivatives

 20.4.7 $\mathop{\theta_{1}\/}\nolimits''(0,q)=\mathop{\theta_{2}\/}\nolimits'''\!\left% (0,q\right)=\mathop{\theta_{3}\/}\nolimits'''\!\left(0,q\right)=\mathop{\theta% _{4}\/}\nolimits'''\!\left(0,q\right)=0.$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $(\NVar{a},\NVar{b})$: open interval and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E7 Encodings: TeX, pMML, png See also: Annotations for 20.4(ii)
 20.4.8 $\displaystyle\frac{\mathop{\theta_{1}\/}\nolimits'''\!\left(0,q\right)}{% \mathop{\theta_{1}\/}\nolimits'\!\left(0,q\right)}$ $\displaystyle=-1+24\sum_{n=1}^{\infty}\frac{q^{2n}}{(1-q^{2n})^{2}}.$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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) 20.4.9 $\displaystyle\frac{\mathop{\theta_{2}\/}\nolimits''\!\left(0,q\right)}{\mathop% {\theta_{2}\/}\nolimits\!\left(0,q\right)}$ $\displaystyle=-1-8\sum_{n=1}^{\infty}\frac{q^{2n}}{(1+q^{2n})^{2}},$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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) 20.4.10 $\displaystyle\frac{\mathop{\theta_{3}\/}\nolimits''\!\left(0,q\right)}{\mathop% {\theta_{3}\/}\nolimits\!\left(0,q\right)}$ $\displaystyle=-8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1+q^{2n-1})^{2}},$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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) 20.4.11 $\displaystyle\frac{\mathop{\theta_{4}\/}\nolimits''\!\left(0,q\right)}{\mathop% {\theta_{4}\/}\nolimits\!\left(0,q\right)}$ $\displaystyle=8\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}.$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: 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)
 20.4.12 $\frac{\mathop{\theta_{1}\/}\nolimits'''\!\left(0,q\right)}{\mathop{\theta_{1}% \/}\nolimits'\!\left(0,q\right)}=\frac{\mathop{\theta_{2}\/}\nolimits''\!\left% (0,q\right)}{\mathop{\theta_{2}\/}\nolimits\!\left(0,q\right)}+\frac{\mathop{% \theta_{3}\/}\nolimits''\!\left(0,q\right)}{\mathop{\theta_{3}\/}\nolimits\!% \left(0,q\right)}+\frac{\mathop{\theta_{4}\/}\nolimits''\!\left(0,q\right)}{% \mathop{\theta_{4}\/}\nolimits\!\left(0,q\right)}.$ Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function and $q$: nome Permalink: http://dlmf.nist.gov/20.4.E12 Encodings: TeX, pMML, png See also: Annotations for 20.4(ii)