Inverse Hyperbolic Cotangent of Imaginary Number

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Theorem

$\map {\coth^{-1} } {i x} = i \cot^{-1} x$


Proof

\(\displaystyle y\) \(=\) \(\displaystyle \map {\coth^{-1} } {i x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \coth y\) \(=\) \(\displaystyle i x\) Definition of Inverse Hyperbolic Cotangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle i \coth y\) \(=\) \(\displaystyle - x\) $i^2 = -1$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \cot {i y}\) \(=\) \(\displaystyle x\) Hyperbolic Cotangent in terms of Cotangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle i y\) \(=\) \(\displaystyle \cot^{-1} x\) Definition of Inverse Cotangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle -i \cot^{-1} x\) multiplying both sides by $-i$

$\blacksquare$


Sources