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
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
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.
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