expi

Expi is not a single, officially defined term in mathematics or science, but in informal usage it is often seen as a shorthand for the complex exponential function evaluated at an imaginary argument, written as exp(i). The exponential function exp(z) is defined for all complex numbers z as e^z.

In mathematics, exp(iθ) is described by Euler's formula: exp(iθ) = cos θ + i sin θ, for any real θ. When θ = 1 (radian), exp(i) equals cos(1) + i sin(1), which is approximately 0.540302 + 0.841471 i. The magnitude of exp(i) is 1, and its argument is 1 radian, placing it on the unit circle in the complex plane.

The exact string "expi" is not a standard mathematical symbol. Proper notation typically uses exp(i) or e^i rather than a combined word. In plain-text notes, some writers may temporarily use "expi" to mean exp(i), but this is not universal and can cause ambiguity. When clarity is required, it is best to write exp(i) or e^i.

Applications of complex exponentials are widespread. They underpin Euler’s formula, enable compact representations of oscillatory functions, and appear in Fourier analysis, signal processing, quantum mechanics, and control theory. Expressions such as e^{iωt} model rotating phasors and propagating waves, illustrating how exponential and trigonometric forms are interconnected in the complex plane.