# Hyperbolic functions

This is an hyperbolic functions calculator that accepts real and complex numbers.
input sqrt(2) for square root of 2 for example.

This tool calculates hyperbolic trigonometric functions: hyperbolic sinus, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent for a given real or complex number.

## 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, +∞ [.