Definition:Inverse Hyperbolic Cosecant/Complex/Principal Branch

Definition

The principal branch of the complex inverse hyperbolic cosecant function is defined as:

$\forall z \in \C_{\ne 0}: \map \Arcsch z := \map \Ln {\dfrac {1 + \sqrt {z^2 + 1} } z}$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm

$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.

Also see

• Derivation of Area Hyperbolic Cosecant from Inverse Hyperbolic Cosecant Multifunction

• Definition:Complex Area Hyperbolic Sine
• Definition:Complex Area Hyperbolic Cosine
• Definition:Complex Area Hyperbolic Tangent
• Definition:Complex Area Hyperbolic Cotangent
• Definition:Complex Area Hyperbolic Secant

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $8$