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 $-i$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.93$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions