Inverse Cotangent of Imaginary Number

Theorem

$\map {\cot^{-1} } {i x} = - i \coth^{-1} x$

$\ds y$

$=$

$\ds \map {\cot^{-1} } {i x}$

$\ds \leadsto \ \ $

$\ds \cot y$

$=$

$\ds i x$

Definition of Inverse Cotangent

$\ds \leadsto \ \ $

$\ds i \cot y$

$=$

$\ds - x$

$i^2 = -1$

$\ds \leadsto \ \ $

$\ds \map \coth {i y}$

$=$

$\ds x$

$\ds \leadsto \ \ $

$\ds i y$

$=$

$\ds \coth^{-1} x$

Definition of Inverse Hyperbolic Cotangent

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds -i \coth^{-1} x$

multiplying both sides by $-i$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.98$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions