Equivalence of Definitions of Complex Inverse Hyperbolic Tangent

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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

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

Definition 1

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

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

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

Definition 2

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

$\forall z \in S: \tanh^{-1} \left({z}\right) := \left\{{\dfrac 1 2 \ln \left({\dfrac {1 + z} {1 - z} }\right) + k \pi i: k \in \Z}\right\}$

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


Proof

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

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


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

\(\displaystyle 1 + z\) \(=\) \(\displaystyle 0 + 0 i\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {1 + z} {1 - z}\) \(=\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle \ln \left({\dfrac {1 + z} {1 - z} }\right)\) \(\) \(\displaystyle \text {is undefined}\)


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

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


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


Definition 1 implies Definition 2

It is demonstrated that:

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


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


Then:

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


Thus by definition of subset:

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

$\Box$


Definition 2 implies Definition 1

It is demonstrated that:

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

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

Then:

\(\displaystyle \exists k \in \Z: \ \ \) \(\displaystyle w\) \(=\) \(\displaystyle \dfrac 1 2 \ln \left({\dfrac {1 + z} {1 - z} }\right) + k \pi i\)
\(\displaystyle \implies \ \ \) \(\displaystyle \exists k \in \Z: \ \ \) \(\displaystyle 2 w + 2 \left({-k}\right) \pi i\) \(=\) \(\displaystyle \ln \left({\dfrac {1 + z} {1 - z} }\right)\)
\(\displaystyle \implies \ \ \) \(\displaystyle e^{2 w + 2 \left({-k}\right) \pi i}\) \(=\) \(\displaystyle \dfrac {1 + z} {1 - z}\) Definition of Complex Natural Logarithm
\(\displaystyle \implies \ \ \) \(\displaystyle e^{2 w}\) \(=\) \(\displaystyle \dfrac {1 + z} {1 - z}\) Complex Exponential Function has Imaginary Period
\(\displaystyle \implies \ \ \) \(\displaystyle z\) \(=\) \(\displaystyle \dfrac {e^w - e^{- w} } {e^w + e^{- w} }\)
\(\displaystyle \implies \ \ \) \(\displaystyle z\) \(=\) \(\displaystyle \tanh w\) Definition of Hyperbolic Tangent
\(\displaystyle \implies \ \ \) \(\displaystyle w\) \(\in\) \(\displaystyle \left\{ {w \in \C: \tanh \left({w}\right) = z}\right\}\)


Thus by definition of superset:

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

$\Box$


Thus by definition of set equality:

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

$\blacksquare$