Inverse Tangent of Imaginary Number/Proof 2
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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 |
$\blacksquare$