# Inverse Hyperbolic Tangent is Odd Function

## Theorem

Let $x \in \R$.

Then:

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

where $\map {\tanh^{-1} } {-x}$ denotes the inverse hyperbolic tangent function.

## Proof 1

 $\displaystyle \map {\tanh^{-1} } {-x}$ $=$ $\displaystyle y$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle -x$ $=$ $\displaystyle \tanh y$ Definition 1 of Inverse Hyperbolic Tangent $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $=$ $\displaystyle -\tanh y$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $=$ $\displaystyle \map \tanh {-y}$ Hyperbolic Tangent Function is Odd $\displaystyle \leadstoandfrom \ \$ $\displaystyle \tanh^{-1} x$ $=$ $\displaystyle -y$ Definition 1 of Inverse Hyperbolic Tangent

$\blacksquare$

## Proof 2

 $\displaystyle \tanh^{-1} \left({-x}\right)$ $=$ $\displaystyle \frac{1}{2}\ln\left(\frac{1+\left(-x\right)}{1-\left(-x\right)}\right)$ Definition of Inverse Hyperbolic Tangent $\displaystyle$ $=$ $\displaystyle \frac{1}{2}\ln \left(\frac{1-x}{1+x}\right)$ $\displaystyle$ $=$ $\displaystyle -\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$ Logarithm of Power $\displaystyle$ $=$ $\displaystyle -\tanh^{-1} \left(x\right)$ Definition of Inverse Hyperbolic Tangent

$\blacksquare$