Digital Library of Mathematical 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

Notes:: Replace by in §4.22.
Keywords:: hyperbolic functions, infinite products, partial fractions
Permalink:: http://dlmf.nist.gov/4.36

4.36.1 $\mathop{\sinh\/}\nolimits z=z\prod_{{n=1}}^{\infty}\left(1+\frac{z^{2}}{n^{2}% \pi^{2}}\right),$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, : integer and : complex variable
A&S Ref:: 4.5.68
Permalink:: http://dlmf.nist.gov/4.36.E1
Encodings:: TeX, pMML, png

4.36.2 $\mathop{\cosh\/}\nolimits z=\prod_{{n=1}}^{\infty}\left(1+\frac{4z^{2}}{(2n-1)% ^{2}\pi^{2}}\right).$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, : integer and : complex variable
A&S Ref:: 4.5.69
Permalink:: http://dlmf.nist.gov/4.36.E2
Encodings:: TeX, pMML, png

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

4.36.3 $\mathop{\coth\/}\nolimits z=\frac{1}{z}+2z\sum_{{n=1}}^{\infty}\frac{1}{z^{2}+% n^{2}\pi^{2}},$

Symbols:: $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/4.36.E3
Encodings:: TeX, pMML, png

4.36.4 ${\mathop{\mathrm{csch}\/}\nolimits^{{2}}}z=\sum_{{n=-\infty}}^{\infty}\frac{1}% {(z-n\pi i)^{2}},$

Symbols:: $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/4.36.E4
Encodings:: TeX, pMML, png

4.36.5 $\mathop{\mathrm{csch}\/}\nolimits z=\frac{1}{z}+2z\sum_{{n=1}}^{\infty}\frac{(% -1)^{n}}{z^{2}+n^{2}\pi^{2}}.$

Symbols:: $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/4.36.E5
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 4.35 Identities 4.37 Inverse Hyperbolic Functions