Hyperbolic functions
input sqrt(2) for square root of 2 for example.
Hyperbolic functions in R (real numbers)
| Function | Abbreviation | Domain | Range |
|---|---|---|---|
| hyperbolic sine | y = sinh(x) | All real numbers | All real numbers |
| hyperbolic cosine | y = cosh(x) | All real numbers | y >= 1 |
| hyperbolic tangent | y = tanh(x) | All real numbers | -1 < y < 1 |
| hyperbolic cotangent | y = coth (x) | all non-zero real numbers | y < -1 ou y > 1 |
hyperbolic sine
The hyperbolic sinus function is defined as follows,
`sinh(x) = (e^x - e^ (-x)) /2`
sinh(x) is defined for all real numbers x so the definition domain is `RR`.
The range is `RR`.
hyperbolic cosine
The hyperbolic cosine function is defined as follows,
`cosh (x) = (e^x + e^ (-x)) /2`
cosh(x) is defined for all real numbers x so the definition domain is `RR`.
The range (set of function values) is [1, +∞[.
hyperbolic tangent
The hyperbolic tangent is defined as the ratio between the hyperbolic sine and the hyperbolic cosine functions.
`tanh (x) = frac {sinh (x)} {cosh (x)} = frac {e^x + e^ (-x)} {e^x + e^ (-x)} = frac {e^ (2x) - 1} {e^ (2x) + 1} `
tanh(x) is defined for all real numbers x so the definition domain is `RR`.
The range (set of function values) is] -1, 1 [.
hyperbolic cotangent
The hyperbolic cotangent is defined as the ratio between the hyperbolic cosine and the hyperbolic sine functions.
`coth (x) = frac {cosh (x)} {sinh (x)} = frac {e^x - e^ (-x)} {e^x - e^ (-x)} = frac {e^ (2x) + 1} {e^ (2x) - 1} `
It can also defined as the hyperbolic tangent reciprocal,
`coth(x) = frac {1} {tanh (x)}`
coth(x) is defined for all non-zero real numbers so the definition domain is the set of nonzero real: `RR`\ {0}.
The range is ] -∞, -1 [ U ] 1, +∞ [.