4 Elementary Functions Hyperbolic Functions 4.32 Inequalities 4.34 Derivatives and Differential Equations

§4.33 Maclaurin Series and Laurent Series

ⓘ

Keywords:: Laurent series, Maclaurin series, hyperbolic functions
Referenced by:: §24.2(i)
Permalink:: http://dlmf.nist.gov/4.33
See also:: Annotations for Ch.4

4.33.1		$\sinh z=z+\frac{z^{3}}{3!}+\frac{z^{5}}{5!}+\cdots,$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable A&S Ref: 4.5.62 Permalink: http://dlmf.nist.gov/4.33.E1 Encodings: TeX, pMML, png See also: Annotations for §4.33 and Ch.4

4.33.2		$\cosh z=1+\frac{z^{2}}{2!}+\frac{z^{4}}{4!}+\cdots.$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable A&S Ref: 4.5.63 Permalink: http://dlmf.nist.gov/4.33.E2 Encodings: TeX, pMML, png See also: Annotations for §4.33 and Ch.4

4.33.3		$\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$
$\|z\|<\frac{1}{2}\pi$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tanh\NVar{z}$ : hyperbolic tangent function, $n$ : integer and $z$ : complex variable A&S Ref: 4.5.64 Permalink: http://dlmf.nist.gov/4.33.E3 Encodings: TeX, pMML, png See also: Annotations for §4.33 and Ch.4

For $B_{2n}$ see §24.2(i). For expansions that correspond to (4.19.4)–(4.19.9), change $z$ to $i z$ and use (4.28.8)–(4.28.13).

4.32 Inequalities 4.34 Derivatives and Differential Equations

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