I know that the imaginary part of $\tanh^{-1}(x)$ is equal to $-i \pi/2$ for $x>1$. I am wondering if there is a known explicit expression for the real part?
2026-03-27 16:26:15.1774628775
Expression for real part of ArcTanh(x) for x > 1
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I think I found the answer by inspection :
$$ \tanh^{-1}(x) = - i \frac{\pi}{2} + \tanh^{-1}(1/x) $$ for $x>1$.