# 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:

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

$\blacksquare$