Equivalence of Definitions of Complex Inverse Cosine Function

Theorem

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

Definition 1

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

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

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

where $\cos \left({w}\right)$ is the cosine of $w$.

Definition 2

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

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

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

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 $z \in \C$:

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

Thus let $z \in \C$.

Definition 1 implies Definition 2

It will be demonstrated that:

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

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

From Euler's Cosine Identity:

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

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

Then:

\(\ds 2 z\)

\(=\)

\(\ds v + \frac 1 v\)

multiplying $(1)$ by $2$

\(\ds \leadsto \ \ \)

\(\ds v^2 - 2 z v + 1\)

\(=\)

\(\ds 0\)

multiplying by $v$ and rearranging

\(\ds \leadsto \ \ \)

\(\ds v\)

\(=\)

\(\ds z + \paren {z^2 - 1}^{1/2}\)

Quadratic Formula

Let $s = z^2 - 1$.

Then:

\(\ds v\)

\(=\)

\(\ds z + s^{1/2}\)

\(\ds \)

\(=\)

\(\ds z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} }\)

Definition of Complex Square Root

\(\text {(2)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds \ln v\)

\(=\)

\(\ds \map \ln {z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} } }\)

where $\ln$ denotes the Complex Natural Logarithm

We have that:

\(\ds v\)

\(=\)

\(\ds e^{i w}\)

\(\ds \leadsto \ \ \)

\(\ds \ln v\)

\(=\)

\(\ds \map \ln {e^{i w} }\)

\(\text {(3)}: \quad\)

\(\ds \exists k' \in \Z: \, \)

\(\ds \)

\(=\)

\(\ds i w + 2 k' \pi i\)

Definition of Complex Natural Logarithm

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

\(\ds i w + 2 k' \pi i\)

\(=\)

\(\ds \map \ln {z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} } }\)

\(\ds \leadsto \ \ \)

\(\ds w\)

\(=\)

\(\ds \frac 1 i \map \ln {z + \sqrt {\cmod s} \paren {\map \cos {\frac {\map \arg s} 2} + i \map \sin {\frac {\map \arg s} 2} } } + 2 k \pi\)

putting $k = -k'$

\(\ds \leadsto \ \ \)

\(\ds w\)

\(=\)

\(\ds \frac 1 i \map \ln {z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } } + 2 k \pi\)

Definition of Exponential Form of Complex Number

Thus by definition of subset:

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

$\Box$

Definition 2 implies Definition 1

It will be demonstrated that:

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

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

Then:

\(\ds \exists k \in \Z: \, \)

\(\ds i w + 2 \paren {-k} \pi i\)

\(=\)

\(\ds \map \ln {z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } }\)

\(\ds \leadsto \ \ \)

\(\ds e^{i w + 2 \paren {-k} \pi i}\)

\(=\)

\(\ds z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} }\)

Definition of Complex Natural Logarithm

\(\ds \leadsto \ \ \)

\(\ds e^{i w}\)

\(=\)

\(\ds z + \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} }\)

Complex Exponential Function has Imaginary Period

\(\ds \leadsto \ \ \)

\(\ds e^{i w} - z\)

\(=\)

\(\ds \sqrt {\cmod {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} }\)

\(\ds \leadsto \ \ \)

\(\ds \paren {e^{i w} - z}^2\)

\(=\)

\(\ds \cmod {z^2 - 1} e^{i \map \arg {z^2 - 1} }\)

Roots of Complex Number

\(\ds \leadsto \ \ \)

\(\ds \paren {e^{i w} - z}^2\)

\(=\)

\(\ds z^2 - 1\)

Definition of Exponential Form of Complex Number

\(\ds \leadsto \ \ \)

\(\ds e^{2 i w} - 2 z e^{i w} + z^2\)

\(=\)

\(\ds z^2 - 1\)

Square of Difference

\(\ds \leadsto \ \ \)

\(\ds e^{2 i w} + 1\)

\(=\)

\(\ds 2 z e^{i w}\)

\(\ds \leadsto \ \ \)

\(\ds e^{i w} + \frac 1 {e^{i w} }\)

\(=\)

\(\ds 2 z\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds \frac {e^{i w} + e^{-i w} } 2\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds \cos w\)

Euler's Cosine Identity

\(\ds \leadsto \ \ \)

\(\ds w\)

\(\in\)

\(\ds \set {w \in \C: z = \cos w}\)

Thus by definition of superset:

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

$\Box$

Thus by definition of set equality:

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

$\blacksquare$