﻿ arcsech – Algosim documentation
Algosim documentation: arcsech

# arcsech

The inverse hyperbolic secant function.

## Syntax

• `arcsech(x)`

• `x` is a real or complex number

## Description `arcsech` is the inverse of the restriction of the hyperbolic secant function to [0, ∞). For real numbers, the domain is (0, 1]. It is defined by

```arcsech(z) = ln(1/z + √(1/z + 1) ⋅ √(1/z − 1))
```

for all complex `z`. This formula is also used for real numbers outside [0, 1].

## Notes

This function is also called `arsech` (area hyperbolic secant) in the literature. Some authors claim that the name `arcsech` is a misnomer, but this depends crucially on how you view the connections between the trigonometric and the hyperbolic functions and their (restriction) inverses. One can argue that the name `arcsech` makes perfect sense.

## Examples

`arcsech(1/e)`
`1.65745445415`
`arcsech(1)`
`0`
`arcsech(2)`
`1.0471975512⋅i`
`arcsech(−1)`
`3.14159265359⋅i`
`arcsech(i)`
`0.88137358702 − 1.57079632679⋅i`