Expjx

Expjx is a term that may appear in informal texts to denote the complex exponential e^{j x}, where j represents the imaginary unit. The letter j is a convention common in engineering, while i is typically used in mathematics. The expression e^{j x} lies at the heart of Euler's formula, e^{j x} = cos x + i sin x. Consequently, expjx, when interpreted as e^{j x}, lies on the unit circle in the complex plane and has magnitude 1 for real x, with phase equal to x modulo 2π.

Properties of expjx include magnitude 1 and a periodic behavior with period 2π. In applications such as Fourier analysis and signal processing, e^{j x} represents a rotating vector with frequency x (in radians per unit). It is used to decompose signals into sums of complex exponentials and to model sinusoidal components as phasors.

Notation wise, the concatenated form expjx is not standard notation in formal literature; most texts write exp(jx) or e^{j x}. In plain text, exp(jx) or exp(j*x) is commonly used, with parentheses around the argument often required by programming languages or typesetting systems.

See also: Euler's formula, complex exponential, Fourier transform, phasor theory. The concept is rooted in the work of Euler and is foundational in fields that analyze periodic signals and linear systems.