Fourier Series and Convergence: Explained with Examples

A Fourier series expands a periodic function as a sum of sines and cosines: f(x) = a0/2 + Σn=1∞ (ancos nx + bnsin nx). Even a jagged square wave or sawtooth can be built this way — out of perfectly smooth pure tones. The coefficients an, bn measure how much of each frequency the function contains.

This decomposition is the foundation of signal processing and large parts of physics. Engineers filter noise by deleting unwanted frequencies, audio codecs store sounds as frequency data, and the heat equation was solved historically by exactly this expansion — Fourier invented the series to describe heat flow.

Key Concepts

• Orthogonality: on [−π, π], the integral of cos(nx)cos(mx) or sin(nx)sin(mx) is zero for n ≠ m — frequencies do not interfere, so each coefficient is extracted independently.
• Euler formulas: an = (1/π)∫−ππ f(x)cos(nx)dx, bn = (1/π)∫−ππ f(x)sin(nx)dx, a0/2 being the average value.
• Even/odd shortcuts: even functions need only cosines (bn = 0); odd functions need only sines (an = 0) — halving the work.
• Dirichlet's theorem: at points where f is continuous the series converges to f(x); at a jump it converges to the midpoint of the jump.
• Gibbs phenomenon: near a jump, partial sums overshoot by about 9% of the jump height — the overshoot narrows but never disappears as terms are added.

Formulas

• f(x) = a0/2 + Σn=1∞ (ancos nx + bnsin nx).
• a0 = (1/π)∫−ππ f(x)dx; an = (1/π)∫−ππ f(x)cos(nx)dx; bn = (1/π)∫−ππ f(x)sin(nx)dx.
• For period 2L: f(x) = a0/2 + Σ (ancos(nπx/L) + bnsin(nπx/L)) with integrals over [−L, L] scaled by 1/L.
• Assumption: f piecewise smooth. Limit: convergence is pointwise per Dirichlet, not uniform near jumps (Gibbs).

Worked Example

Expand f(x) = x on (−π, π). Since x is odd, all an = 0. Compute bn = (2/π)∫0π x·sin(nx)dx. Integrating by parts: [−x·cos(nx)/n]0π + ∫0π cos(nx)/n dx = −π(−1)n/n + 0. Thus bn = (2/π)(−π(−1)n/n) = 2(−1)n+1/n, and x = Σn=1∞ [2(−1)n+1/n]·sin(nx). Test at x = π/2: π/2 = 2(1 − 1/3 + 1/5 − 1/7 + …), i.e. π/4 = 1 − 1/3 + 1/5 − … — the famous Leibniz series, confirming the expansion. At x = ±π the series sums to 0, the midpoint of the jump in the periodic extension, exactly as Dirichlet predicts.

Common Misconceptions

• The series equals f(x) everywhere. At jump discontinuities it converges to the midpoint of the jump, not to either side's value.
• More terms remove the Gibbs overshoot. Adding terms squeezes the overshoot closer to the jump but its ~9% height persists forever.
• Only periodic functions have Fourier series. Any function on an interval gets a series representing its periodic extension — the sawtooth above is x on (−π, π), repeated.

Real-World Applications

• Solving the heat and wave equations by expanding initial data in sine series.
• Audio synthesis, equalisers, and MP3-style compression operate on frequency content.
• MRI scanners reconstruct images from Fourier-encoded measurements of the body.

A Fourier series writes a periodic function as a0/2 + Σ(ancos nx + bnsin nx), with coefficients from the Euler integral formulas and even/odd symmetry halving the labour. Dirichlet's theorem guarantees convergence to f at continuity points and to the jump midpoint elsewhere, while the Gibbs phenomenon keeps a ~9% overshoot near jumps.