Equivalence of Definitions of Complex Inverse Hyperbolic Cosecant

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Theorem

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

Definition 1

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$

where $\map \csch w$ is the hyperbolic cosecant function.

Definition 2

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

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 show that for all $z \in \C$:

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


Thus let $z \in \C$.


Definition 1 implies Definition 2

It will be demonstrated that:

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


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

From the definition of hyperbolic cosecant:

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


Let $v = e^w$.

Then:

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


Let $s = z^2 + 1$.

Then:

\(\ds v\) \(=\) \(\ds \frac {1 + s^{1/2} } z\)
\(\ds \) \(=\) \(\ds \frac {1 + \sqrt {\cmod s} \paren {\map \cos {\dfrac {\map \arg s} 2} + i \, \map \sin {\dfrac {\map \arg s} 2} } } z\) Definition of Complex Square Root
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \ln v\) \(=\) \(\ds \map \ln {\frac {1 + \sqrt {\cmod s} \paren {\map \cos {\dfrac {\map \arg s} 2} + i \, \map \sin {\dfrac {\map \arg s} 2} } } z}\) 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 $(3)$:

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


Thus by definition of subset:

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

$\Box$


Definition 2 implies Definition 1

It will be demonstrated that:

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


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

Then:

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


Thus by definition of superset:

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

$\Box$


Thus by definition of set equality:

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

$\blacksquare$