4 Elementary Functions Hyperbolic Functions 4.29 Graphics 4.31 Special Values and Limits

§4.30 Elementary Properties

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Keywords:: elementary properties, hyperbolic functions
Notes:: See Hobson (1928, pp. 323–326).
Permalink:: http://dlmf.nist.gov/4.30
See also:: Annotations for Ch.4

Table 4.30.1: Hyperbolic functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

	$\sinh\theta=a$	$\cosh\theta=a$	$\tanh\theta=a$	$\operatorname{csch}\theta=a$	$\operatorname{sech}\theta=a$	$\coth\theta=a$
$\sinh\theta$	$a$	$(a^{2}-1)^{1/2}$	$a(1-a^{2})^{-1/2}$	$a^{-1}$	$a^{-1}(1-a^{2})^{1/2}$	$(a^{2}-1)^{-1/2}$
$\cosh\theta$	$(1+a^{2})^{1/2}$	$a$	$(1-a^{2})^{-1/2}$	$a^{-1}(1+a^{2})^{1/2}$	$a^{-1}$	$a(a^{2}-1)^{-1/2}$
$\tanh\theta$	$a(1+a^{2})^{-1/2}$	$a^{-1}(a^{2}-1)^{1/2}$	$a$	$(1+a^{2})^{-1/2}$	$(1-a^{2})^{1/2}$	$a^{-1}$
$\operatorname{csch}\theta$	$a^{-1}$	$(a^{2}-1)^{-1/2}$	$a^{-1}(1-a^{2})^{1/2}$	$a$	$a(1-a^{2})^{-1/2}$	$(a^{2}-1)^{1/2}$
$\operatorname{sech}\theta$	$(1+a^{2})^{-1/2}$	$a^{-1}$	$(1-a^{2})^{1/2}$	$a(1+a^{2})^{-1/2}$	$a$	$a^{-1}(a^{2}-1)^{1/2}$
$\coth\theta$	$a^{-1}(a^{2}+1)^{1/2}$	$a(a^{2}-1)^{-1/2}$	$a^{-1}$	$(1+a^{2})^{1/2}$	$(1-a^{2})^{-1/2}$	$a$

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Symbols:: $\operatorname{csch}\NVar{z}$ : 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 and $a$ : real or complex constant
A&S Ref:: 4.5.60
Referenced by:: §4.37(vi)
Permalink:: http://dlmf.nist.gov/4.30.T1
See also:: Annotations for §4.30 and Ch.4

4.29 Graphics 4.31 Special Values and Limits

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§4.30 Elementary Properties

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