Inverse Hyperbolic Cotangent is Odd Function

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Theorem

Let $x \in \R$.

Then:

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

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


Proof 1

\(\ds \map {\coth^{-1} } {-x}\) \(=\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds -x\) \(=\) \(\ds \coth y\) Definition 1 of Inverse Hyperbolic Cotangent
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds -\coth y\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds \map \coth {-y}\) Hyperbolic Cotangent Function is Odd
\(\ds \leadstoandfrom \ \ \) \(\ds \coth^{-1} x\) \(=\) \(\ds -y\) Definition 1 of Inverse Hyperbolic Cotangent

$\blacksquare$


Proof 2

\(\ds \map {\coth^{-1} } {-x}\) \(=\) \(\ds \frac 1 2 \map \ln {\frac {-z + 1} {-z - 1} }\) Definition 2 of Inverse Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {\frac {z - 1} {z + 1} }\) multiplying the argument by $\dfrac {-1} {-1}$
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\map \ln {z - 1} - \map \ln {z + 1} }\) Difference of Logarithms
\(\ds \) \(=\) \(\ds -\frac 1 2 \paren {\map \ln {z + 1} - \map \ln {z - 1} }\)
\(\ds \) \(=\) \(\ds -\frac 1 2 \map \ln {\frac {z + 1} {z - 1} }\) Difference of Logarithms
\(\ds \) \(=\) \(\ds -\coth^{-1} x\) Definition 2 of Inverse Hyperbolic Cotangent

$\blacksquare$


Sources

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