# 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

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

$\blacksquare$

## Proof 2

 $\ds \map {\tanh^{-1} } {-x}$ $=$ $\ds \frac 1 2 \map \ln {\frac {1 + \paren {-x} } {1 - \paren {-x} } }$ Definition 2 of Inverse Hyperbolic Tangent $\ds$ $=$ $\ds \frac 1 2 \map \ln {\frac {1 - x} {1 + x} }$ $\ds$ $=$ $\ds \frac 1 2 \paren {\map \ln {1 - x} - \map \ln {1 + x} }$ Difference of Logarithms $\ds$ $=$ $\ds -\frac 1 2 \paren {\map \ln {1 + x} - \map \ln {1 - x} }$ $\ds$ $=$ $\ds -\frac 1 2 \map \ln {\frac {1 + x} {1 - x} }$ Difference of Logarithms $\ds$ $=$ $\ds -\tanh^{-1} x$ Definition 2 of Inverse Hyperbolic Tangent

$\blacksquare$