cosh

Cosh, or the hyperbolic cosine, is a mathematical function defined for all real numbers by cosh x = (e^x + e^{-x})/2. It is an even function, with domain all real numbers and range [1, ∞). As one of the hyperbolic trigonometric functions, cosh x arises from exponential expressions and has several standard identities and properties.

Cosh x can be expanded as a Taylor series cosh x = sum_{n=0}^∞ x^{2n}/(2n)!, and it is related to the hyperbolic sine function sinh x = (e^x − e^{-x})/2. They satisfy the fundamental identity cosh^2 x − sinh^2 x = 1, and the derivatives follow a simple cycle: d/dx cosh x = sinh x and d/dx sinh x = cosh x; hence the second derivative of cosh is cosh itself. The integral of cosh x is sinh x + C.

Analytically, cosh and sinh relate to the ordinary trigonometric functions by analytic continuation: cosh(i x) = cos x and sinh(i x) = i sin x. CosH is a solution to the differential equation y'' = y; general solutions are y = a cosh x + b sinh x, with specific initial conditions selecting particular combinations, such as y(0) = 1, y'(0) = 0 yielding y = cosh x.

Graphically, cosh is U-shaped and symmetric about the y-axis, attaining its minimum value 1 at x = 0. For large |x|, cosh x grows approximately like e^{|x|}. Applications include modeling catenaries in architecture and physics, appearances in hyperbolic geometry and special relativity, and roles in various areas of analysis and probability.