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How does one prove that it is?

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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
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1 Answer

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

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