Fourier Transform
An integral transform that decomposes a continuous time or spatial signal into its constituent frequency or wavevector spectrum.
Governing formula
f̂(k) = ∫_(-∞)^(∞) f(x) e^(-i k x) dx, f(x) = (1 / 2π) ∫_(-∞)^(∞) f̂(k) e^(i k x) dk
SI unit
Inverse Domain Unit (e.g. s⁻¹ or m⁻¹)
In depth
The Fourier transform maps functions between conjugate domains (time to frequency, or position x to wavevector k). It converts differential operators into simple multiplication operators (d/dx → i k), making it indispensable for solving wave equations, optics diffraction, and quantum wave mechanics.
Examples in the real world
Converting a time-domain sound signal into an audio frequency spectrum, or analyzing X-ray diffraction patterns.