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

§4.33 Maclaurin Series and Laurent Series

Keywords:: Laurent series, hyperbolic functions
Referenced by:: §24.2(i)
Permalink:: http://dlmf.nist.gov/4.33

4.33.1 $\mathop{\sinh\/}\nolimits z=z+\frac{z^{3}}{3!}+\frac{z^{5}}{5!}+\cdots,$

Symbols:: : : factorial, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
A&S Ref:: 4.5.62
Permalink:: http://dlmf.nist.gov/4.33.E1
Encodings:: TeX, pMML, png

4.33.2 $\mathop{\cosh\/}\nolimits z=1+\frac{z^{2}}{2!}+\frac{z^{4}}{4!}+\cdots.$

Symbols:: : : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and : complex variable
A&S Ref:: 4.5.63
Permalink:: http://dlmf.nist.gov/4.33.E2
Encodings:: TeX, pMML, png

4.33.3 $\mathop{\tanh\/}\nolimits z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}% z^{7}+\cdots+\frac{2^{{2n}}(2^{{2n}}-1)\mathop{B_{{2n}}\/}\nolimits}{(2n)!}z^{% {2n-1}}+\cdots,$ $|z|<\frac{1}{2}\pi$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : : factorial, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, : integer and : complex variable
A&S Ref:: 4.5.64
Permalink:: http://dlmf.nist.gov/4.33.E3
Encodings:: TeX, pMML, png

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

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