# §23.8 Trigonometric Series and Products

## §23.8(i) Fourier Series

If $q=e^{i\pi\omega_{3}/\omega_{1}}$, $\Im{(z/\omega_{1})}<2\Im{(\omega_{3}/\omega_{1})}$, and $z\notin\mathbb{L}$, then

 23.8.1 $\displaystyle\mathop{\wp\/}\nolimits\!\left(z\right)+\frac{\eta_{1}}{\omega_{1% }}-\frac{\pi^{2}}{4\omega_{1}^{2}}{\mathop{\csc\/}\nolimits^{2}}\!\left(\frac{% \pi z}{2\omega_{1}}\right)$ $\displaystyle=-\frac{2\pi^{2}}{\omega_{1}^{2}}\sum_{n=1}^{\infty}\frac{nq^{2n}% }{1-q^{2n}}\mathop{\cos\/}\nolimits\!\left(\frac{n\pi z}{\omega_{1}}\right),$ 23.8.2 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(z\right)-\frac{\eta_{1}z}{\omega% _{1}}-\frac{\pi}{2\omega_{1}}\mathop{\cot\/}\nolimits\!\left(\frac{\pi z}{2% \omega_{1}}\right)$ $\displaystyle=\frac{2\pi}{\omega_{1}}\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}% }\mathop{\sin\/}\nolimits\!\left(\frac{n\pi z}{\omega_{1}}\right).$

## §23.8(ii) Series of Cosecants and Cotangents

When $z\notin\mathbb{L}$,

 23.8.3 $\mathop{\wp\/}\nolimits\!\left(z\right)=-\frac{\eta_{1}}{\omega_{1}}+\frac{\pi% ^{2}}{4\omega_{1}^{2}}\sum_{n=-\infty}^{\infty}{\mathop{\csc\/}\nolimits^{2}}% \!\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1}}\right),$
 23.8.4 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{\eta_{1}z}{\omega_{1}}+\frac{% \pi}{2\omega_{1}}\sum_{n=-\infty}^{\infty}\mathop{\cot\/}\nolimits\!\left(% \frac{\pi(z+2n\omega_{3})}{2\omega_{1}}\right),$

where in (23.8.4) the terms in $n$ and $-n$ are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)).

 23.8.5 $\eta_{1}=\frac{\pi^{2}}{2\omega_{1}}\left(\frac{1}{6}+\sum_{n=1}^{\infty}{% \mathop{\csc\/}\nolimits^{2}}\!\left(\frac{n\pi\omega_{3}}{\omega_{1}}\right)% \right),$

with similar results for $\eta_{2}$ and $\eta_{3}$ obtainable by use of (23.2.14).

## §23.8(iii) Infinite Products

 23.8.6 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\frac{2\omega_{1}}{\pi}\mathop{\exp% \/}\nolimits\!\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}\right)\mathop{\sin\/}% \nolimits\!\left(\frac{\pi z}{2\omega_{1}}\right)\*\prod_{n=1}^{\infty}\frac{1% -2q^{2n}\mathop{\cos\/}\nolimits\!\left(\pi z/\omega_{1}\right)+q^{4n}}{(1-q^{% 2n})^{2}},$
 23.8.7 $\mathop{\sigma\/}\nolimits\!\left(z\right)=\frac{2\omega_{1}}{\pi}\mathop{\exp% \/}\nolimits\!\left(\frac{\eta_{1}z^{2}}{2\omega_{1}}\right)\mathop{\sin\/}% \nolimits\!\left(\frac{\pi z}{2\omega_{1}}\right)\prod_{n=1}^{\infty}\frac{% \mathop{\sin\/}\nolimits\!\left(\pi(2n\omega_{3}+z)/(2\omega_{1})\right)% \mathop{\sin\/}\nolimits\!\left(\pi(2n\omega_{3}-z)/(2\omega_{1})\right)}{{% \mathop{\sin\/}\nolimits^{2}}\!\left(\pi n\omega_{3}/\omega_{1}\right)}.$