Hyperbolic function

The hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. They are:

(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 y-axis 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 "half-angle 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.