Equivalence of Definitions of Complex Inverse Hyperbolic Tangent

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Theorem

The following definitions of the concept of Complex Inverse Hyperbolic Tangent are equivalent:

Let $S$ be the subset of the complex plane:

$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$

Definition 1

The inverse hyperbolic tangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\tanh^{-1} } z := \set {w \in \C: z = \map \tanh w}$

where $\map \tanh w$ is the hyperbolic tangent function.

Definition 2

The inverse hyperbolic tangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\tanh^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

where $\ln$ denotes the complex natural logarithm considered as a multifunction.


Proof

The proof strategy is to how that for all $z \in S$:

$\set {w \in \C: z = \tanh w} = \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$


Note that when $z = -1 + 0 i$:

\(\ds 1 + z\) \(=\) \(\ds 0 + 0 i\)
\(\ds \leadsto \ \ \) \(\ds \frac {1 + z} {1 - z}\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \map \ln {\dfrac {1 + z} {1 - z} }\) \(\) \(\ds \text {is undefined}\)


Similarly, when $z = 1 + 0 i$:

\(\ds 1 - z\) \(=\) \(\ds 0 + 0 i\)
\(\ds \leadsto \ \ \) \(\ds \frac {1 + z} {1 - z}\) \(\) \(\ds \text {is undefined}\)


Thus let $z \in \C \setminus \set {-1 + 0 i, 1 + 0 i}$.


Definition 1 implies Definition 2

It is demonstrated that:

$\set {w \in \C: z = \tanh w} \subseteq \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$


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


Then:

\(\ds z\) \(=\) \(\ds \frac {e^w - e^{- w} } {e^w + e^{- w} }\) Definition of Hyperbolic Tangent
\(\ds z\) \(=\) \(\ds \frac {e^{2 w} - 1} {e^{2 w} + 1}\) multiplying top and bottom by $e^w$
\(\ds \leadsto \ \ \) \(\ds e^{2 w}\) \(=\) \(\ds \frac {1 + z} {1 - z}\) solving for $e^{2 w}$
\(\ds \leadsto \ \ \) \(\ds \map \ln {e^{2 w} }\) \(=\) \(\ds \ln \frac {1 + z} {1 - z}\)
\(\ds \leadsto \ \ \) \(\ds \exists k' \in \Z: \, \) \(\ds 2 w + 2 k' \pi i\) \(=\) \(\ds \ln \frac {1 + z} {1 - z}\) Definition of Complex Natural Logarithm
\(\ds \leadsto \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds w\) \(=\) \(\ds \frac 1 2 \ln \frac {1 + z} {1 - z} + k \pi i\) putting $k = -k'$


Thus by definition of subset:

$\set {w \in \C: z = \tanh w} \subseteq \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

$\Box$


Definition 2 implies Definition 1

It is demonstrated that:

$\set {w \in \C: z = \tanh w} \supseteq \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

Let $w \in \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$.

Then:

\(\ds \exists k \in \Z: \, \) \(\ds w\) \(=\) \(\ds \dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i\)
\(\ds \leadsto \ \ \) \(\ds \exists k \in \Z: \, \) \(\ds 2 w + 2 \paren {-k} \pi i\) \(=\) \(\ds \map \ln {\dfrac {1 + z} {1 - z} }\)
\(\ds \leadsto \ \ \) \(\ds e^{2 w + 2 \paren {-k} \pi i}\) \(=\) \(\ds \dfrac {1 + z} {1 - z}\) Definition of Complex Natural Logarithm
\(\ds \leadsto \ \ \) \(\ds e^{2 w}\) \(=\) \(\ds \dfrac {1 + z} {1 - z}\) Complex Exponential Function has Imaginary Period
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds \dfrac {e^w - e^{- w} } {e^w + e^{- w} }\)
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds \tanh w\) Definition of Hyperbolic Tangent
\(\ds \leadsto \ \ \) \(\ds w\) \(\in\) \(\ds \set {w \in \C: z = \tanh w}\)


Thus by definition of superset:

$\set {w \in \C: z = \tanh w} \supseteq \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

$\Box$


Thus by definition of set equality:

$\set {w \in \C: z = \tanh w} = \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

$\blacksquare$

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