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DLMF: §4.36 Infinite Products and Partial Fractions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

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4 Elementary Functions Hyperbolic Functions 4.35 Identities 4.37 Inverse Hyperbolic Functions

§4.36 Infinite Products and Partial Fractions

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Keywords:: hyperbolic functions, infinite products, partial fractions
Notes:: Replace $z$ by $i z$ in §4.22.
Permalink:: http://dlmf.nist.gov/4.36
See also:: Annotations for Ch.4

4.36.1		$\displaystyle\sinh z$	$\displaystyle=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sinh\NVar{z}$ : 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 and Ch.4
4.36.2		$\displaystyle\cosh z$	$\displaystyle=\prod_{n=1}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}% \right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : 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 and Ch.4

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

4.36.3		$\displaystyle\coth z$	$\displaystyle=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{1}{z^{2}+n^{2}\pi^{2}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\coth\NVar{z}$ : hyperbolic cotangent function, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.36.E3 Encodings: TeX, pMML, png See also: Annotations for §4.36 and Ch.4
4.36.4		$\displaystyle{\operatorname{csch}}^{2}z$	$\displaystyle=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\mathrm{i}$ : imaginary unit, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.36.E4 Encodings: TeX, pMML, png See also: Annotations for §4.36 and Ch.4
4.36.5		$\displaystyle\operatorname{csch}z$	$\displaystyle=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}+n^{2}\pi^% {2}}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.36.E5 Encodings: TeX, pMML, png See also: Annotations for §4.36 and Ch.4

4.35 Identities 4.37 Inverse Hyperbolic Functions

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