Hyperbolic function

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

 sinh x = (exp(x) - exp(-x))/2    (hyperbolic sine, pronounced "shine" or "sinch")
 cosh x = (exp(x) + exp(-x))/2    (hyperbolic cosine, pronounced "cosh")
 tanh x = sinh(x)/cosh(x)         (hyperbolic tangent, pronounced "tanch")
 coth x = cosh(x)/sinh(x)         (byperbolic cotangent, pronounced "coth")
 sech x = 1/cosh(x)               (byperbolic secant, pronounced "sech")
 csch x = 1/sinh(x)               (hyperbolic cosecant, pronounced "cosech")

Just as the points (cos x, sin x) define a circle, the points (cosh x, sinh x) define a hyperbola because of the formula

(cosh x)² - (sinh x)² = 1.

The parameter x can no longer be interpreted as an angle, though, 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

sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y)

cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y)

and the "half-angle formulas"

cosh(x/2) = √((1 + cosh(x)) / 2)

|sinh(x/2)| = √((cosh(x) – 1) / 2)

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.

The inverse of the hyperbolic functions are

 arcsinh x = ln(x + √(x² + 1))
 arccosh x = ln(x ± √(x² - 1))
 arctanh x = ln(√(1 - x²) / (1 - x))
 arccoth x = ln(√(1 - 1/x²) / (1 - 1/x))
 arcsech x = ln(1/x ± √(1/x² - 1))
 arccsch x = ln(1/x + √(1/x² + 1))

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.