# Arctangent of Imaginary Number

## Theorem

Let $x$ be a real number.

Then:

$\map {\tan^{-1} } {i x} = \dfrac i 2 \map \ln {\dfrac {1 + x} {1 - x} }$

where $\tan$ is the complex tangent function, $\ln$ is the real natural logarithm, and $i$ is the imaginary unit.

## Proof

Let $y = \map {\tan^{-1} } {i x}$.

Let $x = \tanh \theta$, then $\theta = \tanh^{-1} x$.

 $\displaystyle \tan y$ $=$ $\displaystyle i x$ $\displaystyle \tan y$ $=$ $\displaystyle i \tanh \theta$ $\displaystyle \tan y$ $=$ $\displaystyle \map \tan {i \theta}$ Hyperbolic Tangent in terms of Tangent $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle i \theta$ $\displaystyle y$ $=$ $\displaystyle i \tanh^{-1} x$ $\displaystyle y$ $=$ $\displaystyle \frac i 2 \map \ln {\frac {1 + x} {1 - x} }$ Definition of Real Hyperbolic Arctangent

$\blacksquare$