Equivalence of Definitions of Complex Inverse Sine Function

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Theorem

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

Definition 1

Let $z \in \C$ be a complex number.

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

$\sin^{-1} \paren z := \set {w \in \C: \sin \paren w = z}$

where $\sin \paren w$ is the sine of $w$.

Definition 2

Let $z \in \C$ be a complex number.

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

$\sin^{-1} \paren z := \set {\dfrac 1 i \ln \paren {i z + \sqrt {\cmod {1 - z^2} } \exp \paren {\dfrac i 2 \arg \paren {1 - z^2} } } + 2 k \pi: k \in \Z}$

where:

$\sqrt {\cmod {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
$\arg \paren {1 - z^2}$ denotes the argument of $1 - z^2$
$\ln$ is the complex natural logarithm considered as a multifunction.


Proof

The proof strategy is to show that for all $z \in \C$:

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


Thus let $z \in \C$.


Definition 1 implies Definition 2

It will be demonstrated that:

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


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

From Euler's Sine Identity:

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


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

Then:

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


Let $s = 1 - z^2$.

Then:

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


We have that:

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


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

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


Thus by definition of subset:

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

$\Box$


Definition 2 implies Definition 1

It will be demonstrated that:

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

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

Then:

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


Thus by definition of superset:

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

$\Box$


Thus by definition of set equality:

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

$\blacksquare$

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