Inverse Hyperbolic Sine of Imaginary Number

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Theorem

$\map {\sinh^{-1} } {i x} = i \sin^{-1} x$


Proof

\(\ds y\) \(=\) \(\ds \map {\sinh^{-1} } {i x}\)
\(\ds \leadsto \ \ \) \(\ds \sinh y\) \(=\) \(\ds i x\) Definition of Inverse Hyperbolic Sine
\(\ds \leadsto \ \ \) \(\ds i \sinh y\) \(=\) \(\ds - x\) $i^2 = -1$
\(\ds \leadsto \ \ \) \(\ds \map \sin {i y}\) \(=\) \(\ds -x\) Hyperbolic Sine in terms of Sine
\(\ds \leadsto \ \ \) \(\ds i y\) \(=\) \(\ds \map {\sin^{-1} } {-x}\) Definition of Complex Inverse Sine
\(\ds \leadsto \ \ \) \(\ds i y\) \(=\) \(\ds -\sin^{-1} x\) Inverse Sine is Odd Function
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds i \sinh^{-1} x\) multiplying both sides by $-i$

$\blacksquare$


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Investigate the multifunctional nature of Definition:Complex Inverse Sine and similarly for Hyperbolic Sine
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