Inverse Hyperbolic Tangent is Odd Function/Proof 1

Theorem

$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$

$\ds \map {\tanh^{-1} } {-x}$

$=$

$\ds y$

$\ds \leadstoandfrom \ \ $

$\ds -x$

$=$

$\ds \tanh y$

Definition 1 of Inverse Hyperbolic Tangent

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds -\tanh y$

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds \map \tanh {-y}$

$\ds \leadstoandfrom \ \ $

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

$=$

$\ds -y$

Definition 1 of Inverse Hyperbolic Tangent

$\blacksquare$