arctanh_증명

Theorem

$\displaystyle \tanh^{-1}x=\frac12\ln\left(\frac{1+x}{1-x}\right)$
$\displaystyle (-1\lt x\lt 1)$

Proof

$\displaystyle y=\tanh^{-1}x$
$\displaystyle x=\tanh y$
$\displaystyle \frac{e^y-e^{-y}}{e^y+e^{-y}}=x$
$\displaystyle \frac{e^{2y}-1}{e^{2y}+1}=x$
$\displaystyle e^{2y}-1=x(e^{2y}+1)$
$\displaystyle (1-x)e^{2y}=1+x$
$\displaystyle e^{2y}=\frac{1+x}{1-x}$
$\displaystyle 2y=\ln\left(\frac{1+x}{1-x}\right)$