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