Inverse Tangent of Imaginary Number
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Theorem
- $\map {\tan^{-1} } {i x} = i \tanh^{-1} x$
Proof 1
\(\ds y\) | \(=\) | \(\ds \map {\tan^{-1} } {i x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tan y\) | \(=\) | \(\ds i x\) | Definition of Inverse Tangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds i \tan y\) | \(=\) | \(\ds - x\) | $i^2 = -1$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \tanh {i y}\) | \(=\) | \(\ds -x\) | Tangent in terms of Hyperbolic Tangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds i y\) | \(=\) | \(\ds \map {\tanh^{-1} } {-x}\) | Definition of Inverse Hyperbolic Tangent | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds i y\) | \(=\) | \(\ds -\tanh^{-1} x\) | Inverse Hyperbolic Tangent is Odd Function | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds i \tanh^{-1} x\) | multiplying both sides by $-i$ |
$\blacksquare$
Proof 2
\(\ds \map {\tan^{-1} } {i x}\) | \(=\) | \(\ds \frac i 2 \map \ln {\frac {1 - i \paren {i x} } {1 + i \paren {i x} } }\) | Arctangent Logarithmic Formulation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac i 2 \map \ln {\frac {1 + x} {1 - x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds i \tanh^{-1} x\) | Definition of Inverse Hyperbolic Tangent |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.96$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions