Equivalence of Definitions of Complex Inverse Hyperbolic Cosecant

Proof
The proof strategy is to show that for all $z \in \C$:
 * $\set {w \in \C: z = \csch w} = \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

Thus let $z \in \C$.

Definition 1 implies Definition 2
It will be demonstrated that:


 * $\set {w \in \C: z = \csch w} \subseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

Let $w \in \set {w \in \C: z = \csch w}$.

From the definition of hyperbolic cosecant:


 * $(1): \quad z = \dfrac 2 {e^w - e^{- w} }$

Let $v = e^w$.

Then:

Let $s = z^2 + 1$.

Then:

We have that:

Thus from $(2)$ and $(3)$:

Thus by definition of subset:
 * $\set {w \in \C: z = \csch w} \subseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

Definition 2 implies Definition 1
It will be demonstrated that:


 * $\set {w \in \C: z = \csch w} \supseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

Let $w \in \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$.

Then:

Thus by definition of superset:
 * $\set {w \in \C: z = \csch w} \supseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

Thus by definition of set equality:
 * $\set {w \in \C: z = \csch w} = \set {\map \ln {\dfrac {1 + \sqrt {\cmod {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$