Inverse Hyperbolic Cotangent is Odd Function/Proof 2

Theorem

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

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

$=$

$\ds \frac 1 2 \map \ln {\frac {-z + 1} {-z - 1} }$

$\ds $

$=$

$\ds \frac 1 2 \map \ln {\frac {z - 1} {z + 1} }$

multiplying the argument by $\dfrac {-1} {-1}$

$\ds $

$=$

$\ds \frac 1 2 \paren {\map \ln {z - 1} - \map \ln {z + 1} }$

$\ds $

$=$

$\ds -\frac 1 2 \paren {\map \ln {z + 1} - \map \ln {z - 1} }$

$\ds $

$=$

$\ds -\frac 1 2 \map \ln {\frac {z + 1} {z - 1} }$

$\ds $

$=$

$\ds -\coth^{-1} x$

$\blacksquare$