Expikx

Expikx is a shorthand notation commonly used in mathematics and physics to represent the complex exponential function e^{i k x}, where i is the imaginary unit, k is a (real) wave number, and x is a real variable. In higher dimensions, exp(i k · x) extends to a plane wave with wave vector k and position vector x. The form expikx is frequently encountered in Fourier analysis, quantum mechanics, and wave theory as a compact descriptor of monochromatic plane waves.

Notation and variants: The expression is typically written as exp(i k x) or e^{i k x}. In one dimension it reduces to exp(i k x), while in multiple dimensions it becomes exp(i k · x). Some contexts adopt the abbreviated form expikx, particularly in algebraic manipulations or when summarizing families of waves.

Properties: The magnitude of exp(i k x) is unity for all real x, due to Euler’s formula e^{i θ} = cos θ + i sin θ. The derivative with respect to x satisfies d/dx exp(i k x) = i k exp(i k x). For different wave numbers, the integral ∫ exp(i (k − k′) x) dx yields a delta function, reflecting the orthogonality and completeness of plane waves in appropriate domains. Linear combinations of exp(i k x) yield wave packets and are central to Fourier decompositions.

Applications: In physics, exp(i k x) describes plane waves in quantum mechanics, optics, and acoustics, where it encodes phase progression and interference. In mathematics, it forms the basis functions for Fourier series and Fourier transforms, enabling decomposition of signals into frequency components. It also appears as a standard tool for solving linear differential equations with constant coefficients.

History: The utility of complex exponentials originates from Euler’s formula, linking trigonometric and exponential functions. This connection was developed and popularized in the 18th through 20th centuries, becoming foundational in analysis and wave theory.