Equivalence of Definitions of Complex Inverse Hyperbolic Cotangent

Theorem
The following definitions of the complex inverse hyperbolic cotangent are equivalent:

Proof
The proof strategy is to how that for all $z \in \C$:
 * $\left\{{w \in \C: \coth \left({w}\right) = z}\right\} = \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$

Thus let $z \in \C$.

Definition 1 implies Definition 2
It is demonstrated that:


 * $\left\{{w \in \C: \coth \left({w}\right) = z}\right\} \subseteq \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$

Let $w \in \left\{{w \in \C: z = \coth \left({w}\right)}\right\}$.

Then:

Thus by definition of subset:
 * $\left\{{w \in \C: \coth \left({w}\right) = z}\right\} \subseteq \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$

Definition 2 implies Definition 1
It is demonstrated that:


 * $\left\{{w \in \C: \coth \left({w}\right) = z}\right\} \supseteq \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$

Let $w \in \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$.

Then:

Thus by definition of superset:
 * $\left\{{w \in \C: \coth \left({w}\right) = z}\right\} \supseteq \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$

Thus by definition of set equality:
 * $\left\{{w \in \C: \coth \left({w}\right) = z}\right\} = \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$