§4.39 Continued Fractions

Keywords:: continued fractions, hyperbolic functions, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.39

4.39.1 $\mathop{\tanh\/}\nolimits z=\cfrac{z}{1+\cfrac{z^{2}}{3+\cfrac{z^{2}}{5+\cfrac{z^{2}}{7+\cdots}}}},$ $z\neq\pm\tfrac{1}{2}\pi i,\pm\tfrac{3}{2}\pi i,\dots$ .

Symbols:: $\mathop{\tanh\/}\nolimits z$ : 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

4.39.2 $\frac{\mathop{\mathrm{arcsinh}\/}\nolimits z}{\sqrt{1+z^{2}}}=\cfrac{z}{1+\cfrac{1\cdot 2z^{2}}{3+\cfrac{1\cdot 2z^{2}}{5+\cfrac{3\cdot 4z^{2}}{7+\cfrac{3\cdot 4z^{2}}{9+\cdots}}}}},$

Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : 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

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

4.39.3 $\mathop{\mathrm{arctanh}\/}\nolimits z=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{4z^{2}}{5-\cfrac{9z^{2}}{7-\cdots}}}},$

Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : 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

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).