Inverse Hyperbolic Tangent of Imaginary Number
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Theorem
- $\map {\tanh^{-1} } {i x} = i \tan^{-1} x$
Proof
| \(\ds y\) | \(=\) | \(\ds \map {\tanh^{-1} } {i x}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tanh y\) | \(=\) | \(\ds i x\) | Definition of Inverse Hyperbolic Tangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds i \tanh y\) | \(=\) | \(\ds -x\) | $i^2 = -1$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \tan {i y}\) | \(=\) | \(\ds -x\) | Hyperbolic Tangent in terms of Tangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds i y\) | \(=\) | \(\ds \map {\tan^{-1} } {-x}\) | Definition of Inverse Tangent | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds i y\) | \(=\) | \(\ds -\tan^{-1} x\) | Inverse Tangent is Odd Function | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds i \tan^{-1} x\) | multiplying both sides by $-i$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.97$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions