Inverse Hyperbolic Sine of Imaginary Number

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Theorem

$\sinh^{-1} \left({i x}\right) = i \sin^{-1} x$


Proof

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

$\blacksquare$



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