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Hyperbolic Inverse Functions

$\displaystyle \sinh^{-1} z$ $\textstyle =$ $\displaystyle \ln\left({z+\sqrt{z^2+1}\,}\right)$ (1)
$\displaystyle \cosh^{-1} z$ $\textstyle =$ $\displaystyle \ln\left({z\pm\sqrt{z^2-1}\,}\right)$ (2)
$\displaystyle \tanh^{-1} z$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\ln\left({1+z\over 1-z}\right)$ (3)
$\displaystyle \mathop{\rm csch}\nolimits ^{-1} z$ $\textstyle =$ $\displaystyle \ln\left({1\pm \sqrt{1+z^2}\,}\right)$ (4)
$\displaystyle \mathop{\rm sech}\nolimits ^{-1} z$ $\textstyle =$ $\displaystyle \ln\left({1\pm\sqrt{1-z^2}\over z}\right)$ (5)
$\displaystyle \coth^{-1} z$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\ln\left({z+1\over z-1}\right).$ (6)


Abramowitz, M. and Stegun, C. A. (Eds.). ``Hyperbolic Functions.'' §4.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 86-89, 1972.

© 1996-9 Eric W. Weisstein