4 Elementary Functions Hyperbolic Functions 4.38 Inverse Hyperbolic Functions: Further Properties 4.40 Integrals

§4.39 Continued Fractions

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Keywords:: continued fractions, hyperbolic functions, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.39
See also:: Annotations for Ch.4

4.39.1		$\tanh z=\cfrac{z}{1+\cfrac{z^{2}}{3+\cfrac{z^{2}}{5+\cfrac{z^{2}}{7+\cdots}}}},$
$z\neq\pm\tfrac{1}{2}\pi\mathrm{i},\pm\tfrac{3}{2}\pi\mathrm{i},\dots$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit and $z$ : complex variable A&S Ref: 4.5.70 Permalink: http://dlmf.nist.gov/4.39.E1 Encodings: TeX, pMML, png See also: Annotations for §4.39 and Ch.4

4.39.2		$\frac{\operatorname{arcsinh}z}{\sqrt{1+z^{2}}}=\cfrac{z}{1+\cfrac{1\cdot 2z^{2% }}{3+\cfrac{1\cdot 2z^{2}}{5+\cfrac{3\cdot 4z^{2}}{7+\cfrac{3\cdot 4z^{2}}{9+% \cdots}}}}},$
ⓘ Symbols: $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function and $z$ : complex variable A&S Ref: 4.6.36 Permalink: http://dlmf.nist.gov/4.39.E2 Encodings: TeX, pMML, png See also: Annotations for §4.39 and Ch.4

where $z$ is in the open cut plane of Figure 4.37.1(i).

4.39.3		$\operatorname{arctanh}z=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{4z^{2}}{5-\cfrac{9z% ^{2}}{7-\cdots}}}},$
ⓘ Symbols: $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function and $z$ : complex variable A&S Ref: 4.6.35 Permalink: http://dlmf.nist.gov/4.39.E3 Encodings: TeX, pMML, png See also: Annotations for §4.39 and Ch.4

where $z$ is in the open cut plane of Figure 4.37.1(iii).

For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569–571). See also Cuyt et al. (2008, pp. 211–217).

4.38 Inverse Hyperbolic Functions: Further Properties 4.40 Integrals

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§4.39 Continued Fractions

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