Hyperbolics

Hyperbolics is a term often used to refer to the hyperbolic functions, a family of transcendental functions connected to the unit hyperbola and to hyperbolic geometry. The basic functions—sinh, cosh, and tanh—and their reciprocal functions coth, sech, and csch form the core set. They are defined for real arguments via exponential formulas and extend to complex arguments, linking them to trigonometric functions and complex analysis. Hyperbolic functions arise as analogues of circular trigonometric functions but are tied to hyperbolas rather than circles.

The hyperbolic functions are defined by simple exponential expressions: sinh x = (e^x − e^−x)/2, cosh x = (e^x + e^−x)/2, and tanh x = sinh x / cosh x. Key identities include cosh^2 x − sinh^2 x = 1 and the derivatives d/dx sinh x = cosh x, d/dx cosh x = sinh x, and d/dx tanh x = sech^2 x. Addition formulas hold as well: cosh(x+y) = cosh x cosh y + sinh x sinh y, sinh(x+y) = sinh x cosh y + cosh x sinh y, and tanh(x+y) = (tanh x + tanh y)/(1 + tanh x tanh y).

Inverse hyperbolic functions—asinh, acosh, atanh—are defined as the inverse relations of the hyperbolic functions: asinh y = ln(y + sqrt(y^2 + 1)), acosh y = ln(y + sqrt(y^2 − 1)) with y ≥ 1, and atanh y = (1/2) ln((1+y)/(1−y)) for |y| < 1. These functions are useful in integration and in solving certain equations, and they appear in physics, notably in special relativity through the concept of rapidity.

Hyperbolic geometry, a non-Euclidean geometry of constant negative curvature, uses hyperbolic trigonometric identities to describe distances and angles. The hyperbolic functions thus underpin many formulas in geometry, analysis, and physics.