Digital Library of Mathematical Functions
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4 Elementary FunctionsHyperbolic Functions

§4.39 Continued Fractions

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.
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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:
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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}}}}},
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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:
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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}}}},

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).

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