Equivalence of Definitions of Complex Inverse Cosecant Function
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:
- $\set {w \in \C: z = \csc w} = \set {\dfrac 1 i \map \ln {\dfrac {i + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } } z} + 2 k \pi: k \in \Z}$
$\blacksquare$