§4.22 Infinite Products and Partial Fractions

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 $\mathop{\sin\/}\nolimits z=z\prod _{{n=1}}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right),$

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 $\mathop{\cos\/}\nolimits z=\prod _{{n=1}}^{\infty}\left(1-\frac{4z^{2}}{(2n-1)^{2}\pi^{2}}\right).$

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 $\mathop{\cot\/}\nolimits z=\frac{1}{z}+2z\sum _{{n=1}}^{\infty}\frac{1}{z^{2}-n^{2}\pi^{2}},$

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:: TeX, pMML, png

4.22.4 ${\mathop{\csc\/}\nolimits^{{2}}}z=\sum _{{n=-\infty}}^{\infty}\frac{1}{(z-n\pi)^{2}},$

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:: TeX, pMML, png

4.22.5 $\mathop{\csc\/}\nolimits z=\frac{1}{z}+2z\sum _{{n=1}}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}.$

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