# §4.39 Continued Fractions

 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 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 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 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 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).