sinh

Hyperbolic sine, denoted sinh x, is a standard hyperbolic function defined for all real x by sinh x = (e^x − e^(−x))/2. It is an odd function, satisfying sinh(−x) = −sinh(x). It is paired with the hyperbolic cosine cosh x = (e^x + e^(−x))/2, and together they satisfy the identity cosh^2 x − sinh^2 x = 1.

Derivatives and growth: d/dx sinh x = cosh x, and d^2/dx^2 sinh x = sinh x. The function is entire and grows roughly like (1/2) e^{|x|} for large |x|. Its Taylor series around 0 is sinh x = x + x^3/3! + x^5/5! + ... . The inverse function is asinh y (also arsinh), with asinh y = ln(y + sqrt(y^2 + 1)).

Relations and identities: sinh x and cosh x are connected to exponentials by e^x = sinh x + cosh x and e^(−x) = cosh x − sinh x. They participate in numerous hyperbolic identities, such as cosh^2 x − sinh^2 x = 1 and sinh(2x) = 2 sinh x cosh x.

Applications: sinh appears in solving linear differential equations with constant coefficients, in the description of catenary curves via y = a cosh(x/a), and in physics through relativistic concepts like rapidity. It also arises when expressing exponentials of imaginary arguments and in various integral and series contexts.

Overall, sinh x provides a real-valued, smoothly varying bridge between exponential growth and hyperbolic geometry.