§4.31 Special Values and Limits

ⓘ

Table 4.31.1: Hyperbolic functions: values at multiples of

\tfrac{1}{2}\pi i

$z$	0	$\frac{1}{2}\pi\mathrm{i}$	$\pi\mathrm{i}$	$\frac{3}{2}\pi\mathrm{i}$	$\infty$
$\sinh z$	$0$	$\mathrm{i}$	$0$	$-\mathrm{i}$	$\infty$
$\cosh z$	$1$	$0$	$-1$	$0$	$\infty$
$\tanh z$	$0$	$\infty\mathrm{i}$	$0$	$-\infty\mathrm{i}$	$1$
$\operatorname{csch}z$	$\infty$	$-\mathrm{i}$	$\infty$	$\mathrm{i}$	$0$
$\operatorname{sech}z$	$1$	$\infty$	$-1$	$\infty$	$0$
$\coth z$	$\infty$	$0$	$\infty$	$0$	$1$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
A&S Ref:: 4.5.61
Permalink:: http://dlmf.nist.gov/4.31.T1
See also:: Annotations for §4.31 and Ch.4

4.31.1		$\lim_{z\to 0}\frac{\sinh z}{z}=1,$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.31.E1 Encodings: TeX, pMML, png See also: Annotations for §4.31 and Ch.4

4.31.2		$\lim_{z\to 0}\frac{\tanh z}{z}=1,$
ⓘ Symbols: $\tanh\NVar{z}$ : hyperbolic tangent function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.31.E2 Encodings: TeX, pMML, png See also: Annotations for §4.31 and Ch.4

4.31.3		$\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.31.E3 Encodings: TeX, pMML, png See also: Annotations for §4.31 and Ch.4