expx

Expx is the exponential function with base e, commonly written as exp(x) or e^x. The term expx may appear in software and textbooks as a shorthand for applying the exponential function to x. The base e, Euler’s number, is approximately 2.718281828.

Definition and basic properties

Expx maps every real number x to a positive real value. Its domain is all real numbers and its range is (0, ∞). It satisfies expx(0) = 1 and has derivative expx(x) itself, making it the unique function with f'(x) = f(x) and f(0) = 1. Its inverse function is the natural logarithm, ln(x). Fundamental identities include exp(a + b) = exp(a) exp(b) and exp(kx) = (exp(x))^k for real k. In particular, exp(−x) = 1/exp(x).

Series and complex extension

Expx has the power series expansion exp(x) = sum_{n=0}^∞ x^n / n!. For complex numbers z, exp(z) is defined as well and satisfies exp(z) = e^{Re(z)}(cos(Im(z)) + i sin(Im(z))). Euler’s formula gives exp(iθ) = cos θ + i sin θ, linking the exponential to trigonometric functions. The magnitude satisfies |exp(z)| = exp(Re(z)).

Applications

The exponential function models growth and decay, solves differential equations of the form y' = y, and underpins continuous compound interest. It appears in probability distributions, physics, chemistry, and engineering. Computational evaluation uses series, continued fractions, or table-driven methods, with attention to underflow for large negative x and overflow for large positive x.

History

The constant e emerged from problems in compound interest and analysis by mathematicians such as Jacob Bernoulli and Leonhard Euler, who popularized the symbol and notation exp(x).