MAST20029 · Engineering Mathematics
Fourier Series and Fourier Integrals
Fourier methods write a periodic function as a sum of sines and cosines (and a non-periodic one as an integral). You use the Euler formulae for the coefficients, exploit even/odd symmetry (cosine series for even, sine series for odd), and handle functions on (0, L) via odd or even periodic extensions — with the series converging to the midpoint at a jump. Parseval relates energy to the coefficients, and Fourier integrals extend the idea to non-periodic forcing. Examined as Q9–Q10.
What this chapter covers
- 011. Fourier series f(t) = a₀ + Σ [aₙ cos(nωt) + bₙ sin(nωt)], period T = 2L, ω = π/L
- 022. Euler formulae for a₀, aₙ, bₙ as integrals over (−L, L)
- 033. Even functions → cosine series (only a₀, aₙ); odd → sine series (only bₙ)
- 044. Odd and even periodic extensions of a function given on (0, L)
- 055. Convergence at a jump → the midpoint (1/2)[f(t⁻) + f(t⁺)]
- 066. Parseval / energy density relating the mean square to the coefficients
- 077. Application to ODEs with periodic forcing (term-by-term solution)
- 088. Fourier (cosine/sine) integrals for non-periodic functions
Fourier sine series on an interval
- 1 markAn odd extension means only bₙ survive: bₙ = (2/L) ∫₀^L f sin(nωt) dt = (2/2) ∫₀² t sin(nπt/2) dt.
- 1 markIntegrate by parts with u = t, dv = sin(nπt/2) dt: ∫₀² t sin(nπt/2) dt = [−(2t/(nπ)) cos(nπt/2)]₀² + (2/(nπ)) ∫₀² cos(nπt/2) dt.
- 1 markEvaluate: the boundary term is −(4/(nπ)) cos(nπ) = −(4/(nπ))(−1)ⁿ, and the remaining integral is (2/(nπ)) · [(2/(nπ)) sin(nπt/2)]₀² = 0 since sin(nπ) = 0.
- 1 markSo bₙ = −(4/(nπ))(−1)ⁿ = 4(−1)^(n+1)/(nπ).
- 1 markSeries: f(t) = Σ (from n=1 to ∞) [4(−1)^(n+1)/(nπ)] sin(nπt/2).
- 1 markConvergence note: at t = 2 the odd extension has a jump, so the series converges to the midpoint, which is 0.
Key terms
- Fourier series
- A representation f(t) = a₀ + Σ [aₙ cos(nωt) + bₙ sin(nωt)] of a periodic function (period T = 2L, ω = π/L) as a sum of harmonics, with coefficients from the Euler formulae.
- Euler formulae
- The integral formulas for the coefficients: a₀ = (1/2L)∫ f dt, aₙ = (1/L)∫ f cos(nωt) dt, bₙ = (1/L)∫ f sin(nωt) dt, over one period (−L, L).
- Cosine / sine series
- For an even function only a₀ and aₙ appear (cosine series); for an odd function only bₙ appear (sine series), each computed by 2/L times an integral over (0, L).
- Periodic extension
- Extending a function given on (0, L) to all t as an odd extension (for a sine series) or even extension (for a cosine series), then repeating with period 2L.
- Midpoint convergence
- At a jump discontinuity the Fourier series converges not to f but to the midpoint (1/2)[f(t⁻) + f(t⁺)] of the left and right limits.
- Fourier integral
- The non-periodic analogue f(t) = ∫₀^∞ [A(ω) cos ωt + B(ω) sin ωt] dω, with cosine-only for even and sine-only for odd functions; used for non-periodic forcing.
Fourier Series and Fourier Integrals FAQ
When do I get a cosine series versus a sine series?
Symmetry decides. An even function (or an even periodic extension) has only cosine terms — a₀ and aₙ — because the sine integrals vanish. An odd function (or odd extension) has only sine terms bₙ. For a function given on (0, L) you CHOOSE the extension: odd for a sine series, even for a cosine series, which halves the work.
What does the series converge to at a discontinuity?
At a jump the Fourier series converges to the midpoint of the left and right limits, (1/2)[f(t⁻) + f(t⁺)], not to either one-sided value. So at a jump from, say, 2 down to −2, the series gives 0. Stating this midpoint behaviour is a standard exam mark.
How do I sketch a periodic extension?
Take the given function on (0, L). For an odd extension, reflect it through the origin to (−L, 0), then repeat the whole (−L, L) pattern with period 2L; for an even extension, reflect it across the vertical axis instead. Mark the jumps and the midpoint values at the discontinuities — sketching the extension is itself examined.
What is a Fourier integral and when is it used?
It is the continuous analogue of a Fourier series for non-periodic functions: f(t) = ∫₀^∞ [A(ω) cos ωt + B(ω) sin ωt] dω. An even f uses only the cosine integral A(ω); an odd f uses only the sine integral B(ω). It appears when solving ODEs with non-periodic forcing (the Q10 type).
Exam move
Let symmetry do the heavy lifting: decide even (cosine series) or odd (sine series) before computing, so you only evaluate the surviving coefficients over (0, L) with the 2/L factor. Practise the integration-by-parts that the coefficient integrals need, and always finish a piecewise question with the midpoint rule at any jump. Be ready to SKETCH the odd or even periodic extension cleanly, marking jumps and midpoints, since that is directly examined. For Fourier integrals, mirror the same even/odd shortcut with A(ω) or B(ω). Quote the formula you used and keep ω = π/L consistent throughout.
