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4 Elementary Functions Hyperbolic Functions 4.30 Elementary Properties 4.32 Inequalities

§4.31 Special Values and Limits

Notes:: See Hobson (1928, p. 326), Levinson and Redheffer (1970, p. 61).
Keywords:: hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.31

Table 4.31.1: Hyperbolic functions: values at multiples of $\tfrac{1}{2}\pi i$ .

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

Symbols:: $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
A&S Ref:: 4.5.61
Permalink:: http://dlmf.nist.gov/4.31.T1

4.31.1 $\lim_{{z\to 0}}\frac{\mathop{\sinh\/}\nolimits z}{z}=1,$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E1
Encodings:: TeX, pMML, png

4.31.2 $\lim_{{z\to 0}}\frac{\mathop{\tanh\/}\nolimits z}{z}=1,$

Symbols:: $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E2
Encodings:: TeX, pMML, png

4.31.3 $\lim_{{z\to 0}}\frac{\mathop{\cosh\/}\nolimits z-1}{z^{2}}=\frac{1}{2}.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and : complex variable
Permalink:: http://dlmf.nist.gov/4.31.E3
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. 4.30 Elementary Properties 4.32 Inequalities