# §4.22 Infinite Products and Partial Fractions

 4.22.1 $\displaystyle\sin z$ $\displaystyle=z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right),$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function, $n$: integer and $z$: complex variable A&S Ref: 4.3.89 Permalink: http://dlmf.nist.gov/4.22.E1 Encodings: TeX, pMML, png See also: Annotations for §4.22 and Ch.4 4.22.2 $\displaystyle\cos z$ $\displaystyle=\prod_{n=1}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}% \right).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $n$: integer and $z$: complex variable A&S Ref: 4.3.90 Permalink: http://dlmf.nist.gov/4.22.E2 Encodings: TeX, pMML, png See also: Annotations for §4.22 and Ch.4

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

 4.22.3 $\displaystyle\cot z$ $\displaystyle=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$: cotangent function, $n$: integer and $z$: complex variable A&S Ref: 4.3.91 Referenced by: §22.12 Permalink: http://dlmf.nist.gov/4.22.E3 Encodings: TeX, pMML, png See also: Annotations for §4.22 and Ch.4 4.22.4 $\displaystyle{\csc}^{2}z$ $\displaystyle=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi)^{2}},$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$: cosecant function, $n$: integer and $z$: complex variable A&S Ref: 4.3.92 Permalink: http://dlmf.nist.gov/4.22.E4 Encodings: TeX, pMML, png See also: Annotations for §4.22 and Ch.4 4.22.5 $\displaystyle\csc z$ $\displaystyle=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^% {2}}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$: cosecant function, $n$: integer and $z$: complex variable A&S Ref: 4.3.93 Referenced by: §22.12 Permalink: http://dlmf.nist.gov/4.22.E5 Encodings: TeX, pMML, png See also: Annotations for §4.22 and Ch.4