Equivalence of Definitions of Complex Inverse Hyperbolic Sine

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

Thus let $z \in \C$.

Definition 1 implies Definition 2
It is demonstrated that:


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

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

Then by definition of the hyperbolic sine function:


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

Let $v = e^w$.

Then:

Let $s = z^2 + 1$.

Then:

We have that:

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

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

Definition 2 implies Definition 1
It is demonstrated that:


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

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

Then:

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

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