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4 Elementary Functions Trigonometric Functions 4.21 Identities 4.23 Inverse Trigonometric Functions

§4.22 Infinite Products and Partial Fractions

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Keywords:: infinite products, partial fractions, trigonometric functions
Notes:: See Hobson (1928, Chapter 17), Levinson and Redheffer (1970, pp. 387–389).
Referenced by:: §20.5(iii), §4.36
Permalink:: http://dlmf.nist.gov/4.22
See also:: Annotations for Ch.4

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

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