MTH2032 Chap.10 Fourier Series and Half-Range Expansions
Fourier Series and Half-Range Expansions
Week 10 develops Fourier series, the tool that lets separation of variables handle general data. A periodic function satisfying the conditions of the representation theorem equals the sum of its series wherever it is continuous, and the average of its one-sided limits at a jump.
Coefficients come from the Euler formulae; symmetry removes half of them; and half-range expansions turn data on a finite interval into a cosine or sine series through even or odd periodic extension. Each example shows which extension a PDE would need and why.
What this chapter covers
- 01
Periodic functions and fundamental period
- 02
Euler formulae for the coefficients
- 03
Representation theorem and jump values
- 04
Convergence rate and smoothness
- 05
Even and odd functions
- 06
Half-range cosine and sine series
Worked example · free
Sine coefficients of a linear function
- 1Use the odd periodic extension with L = π, so a_0 = a_n = 0 and b_n = (2/π)∫₀^π x sin nx dx.
- 1By parts, ∫₀^π x sin nx dx = −π cos(nπ)/n = π(−1)^(n+1)/n.
- 1So b_n = 2(−1)^(n+1)/n.
- 1Hence x = 2(sin x − sin 2x/2 + sin 3x/3 − …) on (0, π).
Key terms
- Fundamental Period
- The smallest positive period of a periodic function, written T.
- Euler Formulae
- The integral formulas for the coefficients a_0, a_n and b_n of a Fourier series over one period.
- Gibbs Phenomenon
- The persistent overshoot of Fourier partial sums next to a jump discontinuity.
- Even Periodic Extension
- The reflection of a function on [0, L] about the vertical axis, repeated with period 2L.
- Fourier Sine Series
- A Fourier series with only sine terms, produced by an odd function or an odd periodic extension.
- Odd Periodic Extension
- The reflection of a function on [0, L] through the origin, repeated with period 2L, whose series contains only sines.
Fourier Series and Half-Range Expansions FAQ
What value does a Fourier series take at a jump?
By the representation theorem, at a point where f is discontinuous the sum of the series is the mean of the two one-sided limits. Remember that periodicity matters at the ends of a period: the left limit at the start of a period is the value approached at the end of the previous one.
How can I avoid mistakes with T and L?
Write the period T and the half-period L = T/2 at the top of every coefficient calculation. The full-period Euler formulae use 2/T and 2πnx/T, while the half-range formulas use 2/L and nπx/L. The lecture notes describe confusing the two as the most common mistake in this topic.
Why do some Fourier series converge faster than others?
Speed follows smoothness. A periodic function with a jump has coefficients that decrease like 1/n, a continuous one with corners like 1/n², and each extra continuous derivative improves the rate further. An even extension of x on [0, 1] is continuous, so its cosine series converges faster than the sine series of the same data.
Do I need to prove the representation theorem?
The notes say the theorem is not proved in the unit but must be understood and applied: you should check its hypotheses for a given function and use its conclusion to find the value of a series at any point. Proofs of the basic periodicity properties, which take a few lines each, are expected.
Exam move
Start by making integration by parts automatic for x sin nx, x cos nx and x² cos nx, and simplify cos nπ to (−1)^n and sin nπ to zero at once. Then practise full-period series on intervals of your choosing, picking the interval that makes the integrals easiest, as the notes recommend.
Next, drill symmetry: classify functions as even, odd or neither, and use the product rules to discard coefficients before calculating any integral. For half-range expansions, sketch the even and the odd periodic extension for each function before writing a formula; the sketch shows where the jumps are and therefore how fast each series will converge and what value it takes at the ends.
Practise the representation theorem separately, with questions that ask only for the value of the series at given points. Finally, connect every series to Week 11 by asking which boundary conditions would call for it: Dirichlet ends for sine series and Neumann ends for cosine series. Recheck each coefficient by evaluating the first one or two numerically and comparing with a sketch of the function.
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