Inverse Tangent of Imaginary Number/Proof 2

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

\(\ds \map {\tan^{-1} } {i x}\) \(=\) \(\ds \frac i 2 \map \ln {\frac {1 - i \paren {i x} } {1 + i \paren {i x} } }\) Arctangent Logarithmic Formulation
\(\ds \) \(=\) \(\ds \frac i 2 \map \ln {\frac {1 + x} {1 - x} }\)
\(\ds \) \(=\) \(\ds i \tanh^{-1} x\) Definition of Inverse Hyperbolic Tangent

$\blacksquare$