Definition:Inverse Cosecant/Complex/Arccosecant

Definition

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

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

where:

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

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

Symbol

The symbol used to denote the arccosecant function is variously seen as follows:

arccsc

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccosecant function is $\arccsc$.

arccosec

A variant symbol used to denote the arccosecant function is $\operatorname {arccosec}$.

acsc

A variant symbol used to denote the arccosecant function is $\operatorname {acsc}$.

acosec

A variant symbol used to denote the arccosecant function is $\operatorname {acosec}$.

Also see

• Derivation of Complex Arccosecant from Inverse Cosecant Multifunction

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $7$