Equivalence of Definitions of Complex Inverse Hyperbolic Sine

Theorem

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

Definition 1

The inverse hyperbolic sine is a multifunction defined as:

$\forall z \in \C: \map {\sinh^{-1} } z := \set {w \in \C: z = \map \sinh w}$

where $\map \sinh w$ is the hyperbolic sine function.

Definition 2

The inverse hyperbolic sine is a multifunction defined as:

$\forall z \in \C: \map {\sinh^{-1} } z := \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i: k \in \Z}$

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 how that for all $z \in \C$:

$\set {w \in \C: z = \map \sinh w} = \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i}$

Thus let $z \in \C$.

Definition 1 implies Definition 2

It is demonstrated that:

$\set {w \in \C: z = \map \sinh w} \subseteq \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i}$

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

Then by definition of the hyperbolic sine function:

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

Let $v = e^w$.

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

\(\ds \leadsto \ \ \)

\(\ds v\)

\(=\)

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

Quadratic Formula

Let $s = z^2 + 1$.

Then:

\(\ds v\)

\(=\)

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

\(\ds \)

\(=\)

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

Definition of Complex Square Root

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

\(\ds \leadsto \ \ \)

\(\ds \ln v\)

\(=\)

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

where $\ln$ denotes the Complex Natural Logarithm

We have that:

\(\ds v\)

\(=\)

\(\ds e^w\)

\(\ds \leadsto \ \ \)

\(\ds \ln v\)

\(=\)

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

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

\(\ds \)

\(=\)

\(\ds w + 2 k' \pi i: k' \in \Z\)

Definition of Complex Natural Logarithm

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

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

\(=\)

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

\(\ds \leadsto \ \ \)

\(\ds w\)

\(=\)

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

putting $k = -k'$

\(\ds \leadsto \ \ \)

\(\ds w\)

\(=\)

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

Definition of Exponential Form of Complex Number

Thus by definition of subset:

$\set {w \in \C: z = \map \sinh w} \subseteq \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i}$

$\Box$

Definition 2 implies Definition 1

It is demonstrated that:

$\set {w \in \C: z = \map \sinh w} \supseteq \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i}$

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

Then:

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

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

\(=\)

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

\(\ds \leadsto \ \ \)

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

\(=\)

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

Definition of Complex Natural Logarithm

\(\ds \leadsto \ \ \)

\(\ds e^w\)

\(=\)

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

Complex Exponential Function has Imaginary Period

\(\ds \leadsto \ \ \)

\(\ds e^w - z\)

\(=\)

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

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \size {z^2 + 1} e^{i \map \arg {z^2 + 1} }\)

Roots of Complex Number

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds z^2 + 1\)

Definition of Exponential Form of Complex Number

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds z^2 + 1\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds 1 + 2 z e^w\)

\(\ds \leadsto \ \ \)

\(\ds e^w - \frac 1 {e^w}\)

\(=\)

\(\ds 2 z\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

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

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds \sinh w\)

Definition of Hyperbolic Sine

\(\ds \leadsto \ \ \)

\(\ds w\)

\(\in\)

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

Thus by definition of superset:

$\set {w \in \C: z = \map \sinh w} \supseteq \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i}$

$\Box$

Thus by definition of set equality:

$\set {w \in \C: z = \map \sinh w} = \set {\map \ln {z + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } + 2 k \pi i}$

$\blacksquare$

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