Equivalence of Definitions of Real Area Hyperbolic Tangent

Theorem
Let $S$ denote the open real interval:
 * $S := \left({-1 \,.\,.\, 1}\right)$

Definition 1 implies Definition 2
Let $x = \tanh y$.

Then:

Definition 2 implies Definition 1
Let $y = \dfrac {1 + x} {1 - x}$.

Therefore:

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Sine
 * Equivalence of Definitions of Real Inverse Hyperbolic Cosine
 * Equivalence of Definitions of Real Inverse Hyperbolic Cosecant
 * Equivalence of Definitions of Real Inverse Hyperbolic Secant
 * Equivalence of Definitions of Real Inverse Hyperbolic Cotangent