Equivalence of Definitions of Complex Inverse Hyperbolic Secant
Theorem
The following definitions of the concept of Complex Inverse Hyperbolic Secant are equivalent:
Definition 1
The inverse hyperbolic secant is a multifunction defined as:
- $\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {w \in \C: z = \map \sech w}$
where $\map \sech w$ is the hyperbolic secant function.
Definition 2
The inverse hyperbolic secant is a multifunction defined as:
- $\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
where:
- $\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
- $\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$
- $\ln$ denotes the complex natural logarithm as a multifunction.
Proof
The proof strategy is to show that for all $z \in \C_{\ne 0}$:
- $\set {w \in \C: z = \sech w} = \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
Thus let $z \in \C_{\ne 0}$.
Definition 1 implies Definition 2
It will be demonstrated that:
- $\set {w \in \C: z = \sech w} \subseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
Let $w \in \set {w \in \C: z = \sech w}$.
From the definition of hyperbolic secant:
- $(1): \quad z = \dfrac 2 {e^w + e^{- w}}$
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 $s = 1 - z^2$.
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:
\(\ds v\) | \(=\) | \(\ds e^w\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln v\) | \(=\) | \(\ds \map \ln {e^w}\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \exists k' \in \Z: \, \) | \(\ds \ln v\) | \(=\) | \(\ds w + 2 k' \pi i\) | 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 $k = -k'$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds w\) | \(=\) | \(\ds \map \ln {\frac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i\) | Definition of Exponential Form of Complex Number |
Thus by definition of subset:
- $\set {w \in \C: z = \sech w} \subseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
$\Box$
Definition 2 implies Definition 1
It will be demonstrated that:
- $\set {w \in \C: z = \sech w} \supseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
Let $w \in \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$.
Then:
\(\ds \exists k \in \Z: \, \) | \(\ds w + 2 \paren {-k} \pi i\) | \(=\) | \(\ds \map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^{w + 2 \paren {-k} \pi i}\) | \(=\) | \(\ds \dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z\) | Definition of Complex Natural Logarithm | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^w\) | \(=\) | \(\ds \dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z\) | Complex Exponential Function has Imaginary Period | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z e^w - 1\) | \(=\) | \(\ds \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {z e^w - 1}^2\) | \(=\) | \(\ds \cmod {1 - z^2} e^{i \map \arg {1 - z^2} }\) | Roots of Complex Number | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {z e^w - 1}^2\) | \(=\) | \(\ds 1 - z^2\) | Definition of Exponential Form of Complex Number | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^2 e^{2 w} - 2 z e^w + 1\) | \(=\) | \(\ds 1 - z^2\) | Square of Difference | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds z^2 e^{2 w} - 2 z e^w\) | \(=\) | \(\ds - z^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z e^{2 w} + z\) | \(=\) | \(\ds 2 e^w\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z \paren {e^w + \frac 1 {e^w} }\) | \(=\) | \(\ds 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \frac 2 {e^w + e^{- w} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \sech w\) | Definition of Hyperbolic Secant | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds w\) | \(\in\) | \(\ds \set {w \in \C: z = \sech w}\) |
Thus by definition of superset:
- $\set {w \in \C: z = \sech w} \supseteq \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
$\Box$
Thus by definition of set equality:
- $\set {w \in \C: z = \sech w} = \set {\map \ln {\dfrac {1 + \sqrt {\cmod {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
$\blacksquare$