# Definition:Inverse Hyperbolic Secant

## Complex Plane

### 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.

## Real Numbers

Let $S$ denote the half-open real interval:

$S := \hointl 0 1$

### Definition 1

The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \map {\sech^{-1} } x := y \in \R_{\ge 0}: x = \map \sech y$

where $\map \sech y$ denotes the hyperbolic secant function.

### Definition 2

The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \map {\sech^{-1} } x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number.
$\sqrt {1 - x^2}$ denotes the positive square root of $1 - x^2$

Hence for $0 < x < 1$, $\map {\sech^{-1} } x$ has $2$ values.

For $x > 0$ and $x > 1$, $\map {\sech^{-1} } x$ is not defined.

## Principal Branch

### Complex Plane

The principal branch of the complex inverse hyperbolic secant function is defined as:

$\forall z \in \C: \map \Arsech z := \map \Ln {\dfrac {1 + \sqrt {1 - z^2} } z}$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.

### Real Numbers

The principal branch of the real inverse hyperbolic secant function is defined as:

$\forall x \in S: \map \arsech x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number.
$\sqrt {1 - x^2}$ specifically denotes the positive square root of $x^2 - 1$

That is, where $\map \arsech x \ge 0$.

## Also known as

The principal branch of the inverse hyperbolic secant is also known as the area hyperbolic secant, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arcsecant, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic secant.

## Also see

• Results about the inverse hyperbolic secant can be found here.