The Fourier Integral

Fourier Analysis · Axiom Academy

Deriving the Fourier Integral from the Fourier Series as Period Approaches Infinity We begin with the complex exponential form of the Fourier series for a function f(t) with period T. The function is represented as a sum over discrete frequencies ω n = n·ω 0 , where ω 0 = 2π/T is the fundamental frequency. Each coefficient c n captures the amplitude and phase of the nth harmonic component. Notice that the frequencies are evenly spaced by ω 0 . Discrete frequencies: ω n = n·(2π/T) Frequency spacing: Δω = ω 0 = 2π/T As T increases, frequency spacing decreases 2. From Discrete to Continuous Frequencies As we increase the period T, something remarkable happens: the spacing between adjacent frequencies Δω = 2π/T becomes smaller and smaller. In the limit as T → ∞, the discrete set of frequencies becomes a continuous spectrum. We introduce a continuous frequency variable ω, and the discrete spacing Δω becomes the differential element dω. The discrete index n becomes continuous as we transition from a sum to an integral. 3. The Sum Transforms into an Integral When the discrete frequencies become continuous, the sum over discrete harmonics transforms into an integral over all frequencies. We rewrite the Fourier coefficient c n in terms of the continuous variable ω. The key insight: multiply and divide by Δω = 2π/T to prepare for the limit. Define F(ω) = T·c n , so that c n = F(ω)·Δω/(2π). In the limit, the sum becomes an integral.

This is the written version of the interactive lesson above. See the full Fourier Analysis course.