Real Area Hyperbolic Tangent of x over a in Logarithm Form

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Theorem

$\artanh \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} }$


Proof

\(\ds \artanh \frac x a\) \(=\) \(\ds \frac 1 2 \map \ln {\frac {1 + \frac x a} {1 - \frac x a} }\) Definition of Real Area Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac 1 2 \map \ln {\frac {a + x} {a - x} }\) multiplying top and bottom by $a$

$\blacksquare$


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