Digital Library of Mathematical Functions
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4 Elementary FunctionsTrigonometric Functions

§4.22 Infinite Products and Partial Fractions

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Notes:
See Hobson (1928, Chapter 17), Levinson and Redheffer (1970, pp. 387–389).
Keywords:
infinite products, partial fractions, trigonometric functions
Referenced by:
§20.5(iii), §4.36
Permalink:
http://dlmf.nist.gov/4.22
4.22.1\mathop{\sin\/}\nolimits z=z\prod_{{n=1}}^{\infty}\left(1-\frac{z^{2}}{n^{2}% \pi^{2}}\right),
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Symbols:
\mathop{\sin\/}\nolimits 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
4.22.2\mathop{\cos\/}\nolimits z=\prod_{{n=1}}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^% {2}\pi^{2}}\right).
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Symbols:
\mathop{\cos\/}\nolimits 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

When z\neq n\pi, n\in\Integer,

4.22.3\mathop{\cot\/}\nolimits z=\frac{1}{z}+2z\sum_{{n=1}}^{\infty}\frac{1}{z^{2}-n% ^{2}\pi^{2}},
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Symbols:
\mathop{\cot\/}\nolimits 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:
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4.22.4{\mathop{\csc\/}\nolimits^{{2}}}z=\sum_{{n=-\infty}}^{\infty}\frac{1}{(z-n\pi)% ^{2}},
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Symbols:
\mathop{\csc\/}\nolimits z: cosecant function, n: integer and z: complex variable
A&S Ref:
4.3.92
Permalink:
http://dlmf.nist.gov/4.22.E4
Encodings:
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4.22.5\mathop{\csc\/}\nolimits z=\frac{1}{z}+2z\sum_{{n=1}}^{\infty}\frac{(-1)^{n}}{% z^{2}-n^{2}\pi^{2}}.
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Symbols:
\mathop{\csc\/}\nolimits 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
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