# §23.11 Integral Representations

Let $\tau=\ifrac{\omega_{3}}{\omega_{1}}$ and

 23.11.1 $\displaystyle f_{1}(s,\tau)$ $\displaystyle=\frac{{\mathop{\cosh\/}\nolimits^{2}}\!\left(\tfrac{1}{2}\tau s% \right)}{1-2e^{-s}\mathop{\cosh\/}\nolimits\!\left(\tau s\right)+e^{-2s}},$ $\displaystyle f_{2}(s,\tau)$ $\displaystyle=\frac{{\mathop{\cos\/}\nolimits^{2}}\!\left(\tfrac{1}{2}s\right)% }{1-2e^{i\tau s}\mathop{\cos\/}\nolimits s+e^{2i\tau s}}.$

Then

 23.11.2 $\mathop{\wp\/}\nolimits\!\left(z\right)=\frac{1}{z^{2}}+8\int_{0}^{\infty}s% \left(e^{-s}{\mathop{\sinh\/}\nolimits^{2}}\!\left(\tfrac{1}{2}zs\right)f_{1}(% s,\tau)+e^{i\tau s}{\mathop{\sin\/}\nolimits^{2}}\!\left(\tfrac{1}{2}zs\right)% f_{2}(s,\tau)\right)ds,$

and

 23.11.3 $\mathop{\zeta\/}\nolimits\!\left(z\right)=\frac{1}{z}+\int_{0}^{\infty}\left(e% ^{-s}\left(zs-\mathop{\sinh\/}\nolimits\!\left(zs\right)\right)f_{1}(s,\tau)-e% ^{i\tau s}\left(zs-\mathop{\sin\/}\nolimits\!\left(zs\right)\right)f_{2}(s,% \tau)\right)ds,$

provided that $-1<\realpart{(z+\tau)}<1$ and $\left|\imagpart{z}\right|<\imagpart{\tau}$.