Fourier Series

Any periodic function can be written as a sum of sines and cosines. Learn periodic functions, Euler formulas for the coefficients, even and odd symmetry (cosine and sine series), half-range expansions, convergence and the Gibbs phenomenon, Parseval's theorem, and the complex exponential form.

Fourier series decompose square waves from digital clocks, sawtooth sweeps, rectified AC and periodic loads on machinery into harmonics. They are also the key to solving the heat and wave equations by separation of variables.

An analogy

A Fourier series is like mixing paint from primary colours: every periodic shape is a mixture of pure sinusoids in specific amounts (the coefficients).

Harmonics

The fundamental has the signal's period; harmonics are integer multiples. Smooth signals need few harmonics; sharp edges need many.

Symmetry saves work

Check even/odd symmetry first — half the coefficients vanish. Half-wave symmetry (f(x + L) = −f(x)) removes even harmonics.

Half-range expansions

A function on [0, L] can be extended evenly (cosine series) or oddly (sine series) — chosen to match boundary conditions in PDE problems.

Convergence

At jumps the series converges to the midpoint; near jumps it overshoots ≈ 9 % (Gibbs phenomenon) no matter how many terms. Parseval links power in the signal to the sum of harmonic powers.

Formulas

• Series (period 2L): f(x) = a₀/2 + Σ [aₙ cos(nπx/L) + bₙ sin(nπx/L)]
• Coefficients: aₙ = (1/L)∫_{−L}^{L} f cos(nπx/L) dx, bₙ = (1/L)∫_{−L}^{L} f sin(nπx/L) dx
• Symmetry: Even f: bₙ = 0 (cosine series); odd f: aₙ = 0 (sine series)
• Parseval: (1/L)∫ f² dx = a₀²/2 + Σ(aₙ² + bₙ²)
• Complex form: f = Σ cₙ e^(inπx/L), cₙ = (1/2L)∫ f e^(−inπx/L) dx

Worked example: Sawtooth

Given: f(x) = x on (−π, π), period 2π

Find: Coefficients.

• Odd → aₙ = 0
• bₙ = (1/π)∫ x sin nx dx = 2(−1)ⁿ⁺¹/n

Answer: x = 2(sin x − sin 2x/2 + sin 3x/3 − …)

Worked example: Basel sum via Parseval

Given: Sawtooth series

Find: Σ 1/n².

• (1/π)∫x² dx = 2π²/3
• Σ bₙ² = 4Σ1/n² = 2π²/3

Answer: Σ 1/n² = π²/6

Common mistakes

• Wrong: Forgetting a₀/2. — Right: a₀/2 is the average (DC) value.
• Wrong: Wrong period scaling nπx/L. — Right: Use the half-period L consistently.

Applications

• Power electronics: Harmonics of rectifiers and PWM.
• Vibration: Periodic forcing decomposed into harmonics.
• PDEs: Heat and wave equation solutions.

Key takeaways

• Periodic = sum of harmonics.
• Use symmetry to skip coefficients.
• Half-range sine/cosine series for BCs.
• Parseval: power in harmonics.

Going further

• Fourier transform: Non-periodic signals (next lesson).
• Sigma approximation: Reducing Gibbs ripple.
• Orthogonal expansions: Generalised Fourier series (Sturm–Liouville).