Inverse Hyperbolic Sine of Imaginary Number

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Theorem

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


Proof

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

$\blacksquare$



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