Hyperbolic function
The
hyperbolic functions are analogs of the ordinary
trigonometric, or circular, functions. They are:
sinh, cosh and tanh
csch, sech and coth

 (hyperbolic sine, pronounced "shine" or "sinch")

 (hyperbolic cosine, pronounced "cosh")

 (hyperbolic tangent, pronounced "tanch")

 (hyperbolic cotangent, pronounced "coth")

 (hyperbolic secant, pronounced "sech")

 (hyperbolic cosecant, pronounced "cosech")
Relationship to regular trigonometric functions
Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the hyperbola x²  y² = 1 for . This is based on the easily verified identity
 (cosh t)^{2}  (sinh t)^{2} = 1.
However, the parameter
t should not be interpreted as an
angle, and the hyperbolic functions are not
periodic. The function cosh
x is always positive, symmetric with respect to the
yaxis and satisfies cosh 0 = 1, the function sinh
x is symmetric with respect to the origin and hence sinh 0 = 0.
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborne's rule states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of two sinh's. This yields for example the addition theorems


and the "halfangle formulas"

The
derivative of sinh
x is given by cosh
x and the derivative of cosh
x is sinh
x. The graph of the function cosh
x is the
catenary curve.
Inverse hyperbolic functions
The inversess of the hyperbolic functions are





Since the
exponential function can be defined for any
complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh
z and cosh
z are then
holomorphic; their Taylor series expansions are given in the
Taylor series article.