Inverse Hyperbolic Tangent of Imaginary Number

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Theorem

$\map {\tanh^{-1} } {i x} = i \tan^{-1} x$


Proof

\(\displaystyle y\) \(=\) \(\displaystyle \map {\tanh^{-1} } {i x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \tanh y\) \(=\) \(\displaystyle i x\) Definition of Inverse Hyperbolic Tangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle i \tanh y\) \(=\) \(\displaystyle -x\) $i^2 = -1$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \tan {i y}\) \(=\) \(\displaystyle -x\) Hyperbolic Tangent in terms of Tangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle i y\) \(=\) \(\displaystyle \map {\tan^{-1} } {-x}\) Definition of Inverse Tangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle i y\) \(=\) \(\displaystyle -\tan^{-1} x\) Inverse Tangent is Odd Function
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle i \tan^{-1} x\) multiplying both sides by $-i$

$\blacksquare$


Sources