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

§4.30 Elementary Properties

Notes:: See Hobson (1928, pp. 323–326).
Keywords:: hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.30

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

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