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 $a$ |
$\blacksquare$