exp

Exp is the exponential function, typically written as exp(x) = e^x, where e is Euler’s number, approximately 2.71828. It is an elementary function that applies to real and complex arguments, with exp(x) > 0 for all real x. On the real axis its range is (0, ∞). The function is entire in the complex plane, meaning it is analytic everywhere.

Key properties include exp(x + y) = exp(x) exp(y) and d/dx exp(x) = exp(x). It has the power series expansion exp(x) = sum_{n=0}^∞ x^n / n!, and its integral ∫ exp(x) dx = exp(x) + C. The function is the inverse of the natural logarithm, so ln(exp(x)) = x for all real x, and exp(ln x) = x for x > 0. Euler’s formula relates the exponential function to trigonometry via exp(iθ) = cos θ + i sin θ.

Applications of exp are widespread. It is the fundamental solution to the differential equation y′ = y, yielding y = Ce^x, and models continuous growth and decay in science, engineering, and finance (for example, continuous compounding with A = P exp(rt)). In computation, exp is a standard mathematical function in libraries across programming languages, with implementations optimized for accuracy and efficiency.