# Inverse hyperbolic functions

input sqrt(2) for square root of 2 for example.

## Inverse hyperbolic trigonometric functions in R (real numbers)

Function | Abbreviation | Domain | Range |
---|---|---|---|

Inverse hyperbolic sine | y = arsinh(x) | all real numbers | all real numbers |

Inverse hyperbolic cosine | y = arcosh(x) | y >= 1 | y >= 0 |

Inverse hyperbolic tangent | y = artanh(x) | -1 < y < 1 | all real numbers |

Inverse hyperbolic cotangent | y = arcoth(x) | y < -1 ou y > 1 | all non-zero real |

## Inverse hyperbolic sine

Also known as area hyperbolic sine, it is the inverse of the hyperbolic sine function and is defined by,

`\text {arsinh} (x) = ln (x+sqrt (x^2+1))`

arsinh(x) is defined for all real numbers x so the definition domain is `RR`.

The range (set of function values) is `RR`.

## Inverse hyperbolic cosine

Also known as area hyperbolic cosine, it is the inverse of the hyperbolic cosine function and is defined by,

`\text {arcosh} (x) = ln (x+sqrt (x^2-1))`

arcosh(x) is defined for real numbers x, x >= 1 so the definition domain is [1, +∞[.

The range (set of function values) is [0, +∞[.

## Inverse hyperbolic tangent

Also known as area hyperbolic tangent, it is the inverse of the hyperbolic tangent function and is defined by,

`\text {artanh} (x) = frac {1} {2} *ln (frac {1+x} {1-x})`

artanh(x) is defined for real numbers x between -1 and 1 so the definition domain is ]-1, 1[.

The range is the set of real numbers `RR`.

## Inverse hyperbolic cotangent

Also known as area hyperbolic cotangent, it is the inverse of the hyperbolic cotangent function and is defined by,

`\text {arcoth} (x) = frac {1} {2} *ln (frac {x+1} {x-1})`

arcoth(x) is defined for x < -1 ou x > 1 so the definition domain is ] -∞ , -1 [ U ] 1, +∞ [.

The range is the set of non-zero real numbers `RR`\ {0}.