Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How does one prove that it is?

share|cite|improve this question
Hello. You did not say $z$ is a real variable. Perhaps you mean that it is a complex variable? – GEdgar Nov 7 '12 at 18:44
Plug in $z=2$ to get the claim $\mathrm{asinh}(-2\sqrt{-3}/3)=\mathrm{atanh}(2)$, numerically $-.5493061454-1.570796327 i = .5493061443-1.570796327 i$ – GEdgar Nov 7 '12 at 18:47
up vote 5 down vote accepted

Let $\tanh ^{-1}z=y\implies \tanh y=z$

So, $$\frac{\cosh y}1=\frac{\sinh y} z=\frac{\sqrt{\cosh^2 y-\sinh ^2y}}{\sqrt{1-z^2}}=\frac 1{\sqrt{1-z^2}}$$ as $\cosh^2 y-\sinh ^2y=1$

So, $y=\cosh ^{-1}\frac1{\sqrt{1-z^2}}=\sinh ^{-1}\frac z{\sqrt{1-z^2}}$

Or, $$\frac z{\sqrt{1-z^2}}=\frac {\tanh y}{\sqrt{1-\tanh^2 y}}=\frac{\sinh y}{\cosh y\sqrt{1-\frac{\sinh^2 y}{\cosh^2y}}}=\frac{\sinh y}{\sqrt{\cosh^2 y-\sinh ^2y}}=\sinh y$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.