# Inverse Hyperbolic Tangent of Imaginary Number

## 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 $-i$

$\blacksquare$