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Article 18Expansion of Anti-functions.

Since

hence, by integration,

the integration-constant being zero, since ${sinh}^{-1}x$ vanishes with $x$. This series is convergent, and can be used in computation, only when $x<1$. Another series, convergent when $x>1$, is obtained by writing the above derivative in the form

where $C$ is the integration-constant, which will be shown in Art. 19 to be equal to ${log}_{e}2$.

A development of similar form is obtained for ${cosh}^{-1}x$; for

hence

in which $C$ is again equal to ${log}_{e}2$ [Art. 19, Prob. 46]. In order that the function ${cosh}^{-1}x$ may be real, $x$ must not be less than unity; but when $x$ exceeds unity, this series is convergent, hence it is always available for computation.

Again

From (32), (33), (34) are derived:

Prob. 44.
Show that the series for ${tanh}^{-1}x,{coth}^{-1}x,{sech}^{-1}x$, are always available for computation.
Prob. 45.
Show that one or other of the two developments of the inverse hyperbolic cosecant is available.

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