Inverse Hyperbolic Tangent is Odd Function

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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

$\blacksquare$


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