# §4.36 Infinite Products and Partial Fractions

 4.36.1 $\displaystyle\mathop{\sinh\/}\nolimits z$ $\displaystyle=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right),$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function, $n$: integer and $z$: complex variable A&S Ref: 4.5.68 Permalink: http://dlmf.nist.gov/4.36.E1 Encodings: TeX, pMML, png See also: Annotations for 4.36 4.36.2 $\displaystyle\mathop{\cosh\/}\nolimits z$ $\displaystyle=\prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}% \right).$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $n$: integer and $z$: complex variable A&S Ref: 4.5.69 Permalink: http://dlmf.nist.gov/4.36.E2 Encodings: TeX, pMML, png See also: Annotations for 4.36

When $z\neq n\pi i$, $n\in\mathbb{Z}$,

 4.36.3 $\displaystyle\mathop{\coth\/}\nolimits z$ $\displaystyle=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}},$ 4.36.4 $\displaystyle{\mathop{\mathrm{csch}\/}\nolimits^{2}}z$ $\displaystyle=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$ 4.36.5 $\displaystyle\mathop{\mathrm{csch}\/}\nolimits z$ $\displaystyle=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^% {2}}.$