functionssinh

Functionssinh refers to the standard hyperbolic sine function, often represented by sinh. In mathematics and most programming libraries, this function is defined for all real inputs x by sinh(x) = (e^x − e^(−x))/2. It is an odd, strictly increasing function with domain (−∞, ∞) and range (−∞, ∞). It is continuous and differentiable everywhere, with derivative d/dx sinh(x) = cosh(x) and integral ∫ sinh(x) dx = cosh(x) + C.

Key identities include sinh(−x) = −sinh(x) and sinh(x + y) = sinh x cosh y + cosh x sinh y. The hyperbolic cosine satisfies cosh^2 x − sinh^2 x = 1. sinh(x) can be related to exponential and trigonometric functions via sinh(x) = −i sin(ix) or sinh(x) = (e^x − e^(−x))/2.

In numerical computation, functionssinh is typically evaluated via exponentials. For large |x|, direct evaluation can overflow or lose precision; many implementations use stable reformulations or scaling techniques. It also often supports vectorized inputs, applying elementwise to arrays or matrices.

Applications of the hyperbolic sine appear in physics (relativistic velocity expressions), engineering (catenary curves), and the solutions of certain differential equations and hyperbolic geometry problems.

See also asinh, cosh, and exp. Notes: 'functionssinh' is a descriptive name for a function implementing the sinh operation; in standard notation the function is simply called sinh.