Equivalence of Definitions of Real Area Hyperbolic Sine

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

Let $z = e^y$.

Then:

If $x \ge 0$, then:

If $x < 0$, then:

Since the natural logarithm of negative numbers are not defined:
 * $y = \map \ln {x + \sqrt {x^2 + 1} }$

Definition 2 implies Definition 1
Let $y = x + \sqrt {x^2 + 1}$.

Therefore:

Also see

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