The six hyperbolic functions are as follows:
The inverses of these functions are denoted:
NB: The
f-1 notation here implies
inverse function, as opposed to
f raised to the power of -1.
They are defined, in terms of e, in the following way:
ez - e-z
sinh(z) = --------
2
ez + e-z
cosh(z) = --------
2
sinh(z) ez - e-z e2z - 1
tanh(z) = ------- = -------- = --------
cosh(z) ez + e-z e2z + 1
2
sech(z) = --------
ez + e-z
2
csch(z) = --------
ez - e-z
cosh(z) ez + e-z e2z + 1
coth(z) = ------- = -------- = --------
sinh(z) ez - e-z e2z - 1
The following relationships exist between the functions:
sinh(z) = -sinh(-z)
cosh(z) = cosh(-z)
For purely
imaginary arguments,
sinh(iz) = isin(z)
cosh(iz) = cos(z)
Many of the trigonometric identities can be easily transferred to hyperbolic functions, through the use of Osborne's rule.
Some examples are:
cosh2(x) - sinh2(x) = 1
cosh(x) + sinh(x) = ex
cosh(x) - sinh(x) = e-x
For further manipulation of hyperbolic functions one may use the Half Angle Formulae and Double Angle Formulae.