Equivalence of Definitions of Real Area Hyperbolic Cosecant

Definition 1 implies Definition 2
If $x \ge 0$, then:

If $x < 0$, then:

That is:


 * $y = \map \ln {\dfrac 1 x + \dfrac {\sqrt {1 + x^2} } {\size x} }$

Definition 2 implies Definition 1
Therefore:

Also see

 * Equivalence of Definitions of Real Area Hyperbolic Sine
 * Equivalence of Definitions of Real Area Hyperbolic Cosine
 * Equivalence of Definitions of Real Area Hyperbolic Tangent
 * Equivalence of Definitions of Real Area Hyperbolic Cotangent
 * Equivalence of Definitions of Real Area Hyperbolic Secant