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}.