Equivalence of Definitions of Complex Inverse Cosecant Function

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Theorem

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

Definition 1

Let $z \in \C_{\ne 0}$ be a non-zero complex number.

The inverse cosecant of $z$ is the multifunction defined as:

$\csc^{-1} \left({z}\right) := \left\{{w \in \C: \csc \left({w}\right) = z}\right\}$

where $\csc \left({w}\right)$ is the cosecant of $w$.

Definition 2

Let $z \in \C_{\ne 0}$ be a non-zero complex number.

The inverse cosecant of $z$ is the multifunction defined as:

$\csc^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({\dfrac {i + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$

where:

$\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$
$\arg \left({z^2 - 1}\right)$ 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 $z \in \C$:

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


Thus let $z \in \C$.


Definition 1 implies Definition 2

It will be demonstrated that:

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


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

From Euler's Cosecant Identity:

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


Let $v = e^{i w}$.

Then:

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


Let $s = z^2 - 1$.

Then:

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


We have that:

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


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

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


Thus by definition of subset:

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

$\Box$


Definition 2 implies Definition 1

It will be demonstrated that:

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


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

Then:

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


Thus by definition of superset:

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

$\Box$


Thus by definition of set equality:

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

$\blacksquare$