Equivalence of Definitions of Complex Inverse Hyperbolic Cosine

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Theorem

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

Definition 1

The inverse hyperbolic cosine is a multifunction defined as:

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

where $\map \cosh w$ is the hyperbolic cosine function.

Definition 2

The inverse hyperbolic cosine is a multifunction defined as:

$\forall z \in \C: \map {\cosh^{-1} } z := \set {\map \ln {z + \sqrt {\size {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } } + 2 k \pi i: k \in \Z}$

where:

$\sqrt {\size {z^2 - 1} }$ denotes the positive square root of the complex modulus of $z^2 - 1$
$\map \arg {z^2 - 1}$ denotes the argument of $z^2 - 1$
$\ln$ denotes the complex natural logarithm considered as a multifunction.


Proof

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

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


Thus let $z \in \C$.


Definition 1 implies Definition 2

It is demonstrated that:

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


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

By definition of hyperbolic cosine:

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


Let $v = e^w$.

Then:

\(\ds 2 z\) \(=\) \(\ds v + \frac 1 v\) multiplying $(1)$ by $2$
\(\ds \leadsto \ \ \) \(\ds v^2 - 2 z v + 1\) \(=\) \(\ds 0\) multiplying by $v$ and rearranging
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds z + \paren {z^2 - 1}^{1/2}\) Quadratic Formula


Let $s = z^2 - 1$.

Then:

\(\ds v\) \(=\) \(\ds z + s^{1/2}\)
\(\ds \) \(=\) \(\ds z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} }\) Definition of Complex Square Root
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \ln v\) \(=\) \(\ds \map \ln {z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} } }\) where $\ln$ denotes the Complex Natural Logarithm


We have that:

\(\ds v\) \(=\) \(\ds e^w\)
\(\ds \leadsto \ \ \) \(\ds \ln v\) \(=\) \(\ds \map \ln {e^w}\)
\(\text {(3)}: \quad\) \(\ds \exists k' \in \Z: \, \) \(\ds \) \(=\) \(\ds w + 2 k' \pi i\) Definition of Complex Natural Logarithm


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

\(\ds w + 2 k' \pi i\) \(=\) \(\ds \map \ln {z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} } }\)
\(\ds \leadsto \ \ \) \(\ds w\) \(=\) \(\ds \map \ln {z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} } } + 2 k \pi i\) putting $k = -k'$
\(\ds \leadsto \ \ \) \(\ds w\) \(=\) \(\ds \map \ln {z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } } + 2 k \pi i\) Definition of Exponential Form of Complex Number


Thus by definition of subset:

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

$\Box$


Definition 2 implies Definition 1

It is demonstrated that:

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

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

Then:

\(\ds \exists k \in \Z: \, \) \(\ds w + 2 \paren {-k} \pi i\) \(=\) \(\ds \map \ln {z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } }\)
\(\ds \leadsto \ \ \) \(\ds e^{w + 2 \paren {-k} \pi i}\) \(=\) \(\ds z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} }\) Definition of Complex Natural Logarithm
\(\ds \leadsto \ \ \) \(\ds e^w\) \(=\) \(\ds z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} }\) Complex Exponential Function has Imaginary Period
\(\ds \leadsto \ \ \) \(\ds e^w - z\) \(=\) \(\ds \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} }\)
\(\ds \leadsto \ \ \) \(\ds \paren {e^w - z}^2\) \(=\) \(\ds \cmod {z^2 - 1} e^{i \map \arg {z^2 - 1} }\) Roots of Complex Number
\(\ds \leadsto \ \ \) \(\ds \paren {e^w - z}^2\) \(=\) \(\ds z^2 - 1\) Definition of Exponential Form of Complex Number
\(\ds \leadsto \ \ \) \(\ds e^{2 w} - 2 z e^w + z^2\) \(=\) \(\ds z^2 - 1\) Square of Difference
\(\ds \leadsto \ \ \) \(\ds e^{2 w} + 1\) \(=\) \(\ds 2 z e^w\)
\(\ds \leadsto \ \ \) \(\ds e^w + \frac 1 {e^w}\) \(=\) \(\ds 2 z\)
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds \frac {e^w + e^{-w} } 2\)
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds \cosh w\) Definition of Hyperbolic Cosine
\(\ds \leadsto \ \ \) \(\ds w\) \(\in\) \(\ds \set {w \in \C: z = \cosh w}\)


Thus by definition of superset:

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

$\Box$


Thus by definition of set equality:

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

$\blacksquare$