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

\(\ds \map {\tanh^{-1} } {-x}\) \(=\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds -x\) \(=\) \(\ds \tanh y\) 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

$\blacksquare$


Sources

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