MA22S3 Michaelmas 2017

Fourier Analysis for Science

Prof. Ruth Britto [e-mail]

TA James Nelson [e-mail]

TA David Tims [e-mail]

Lectures:

• Tuesdays, 1 pm, Joly

• Thursdays, 4 pm, Lloyd Building 04

• Fridays, 1 pm, Lloyd Building 11

Tutorials take place on Wednesdays, Thursdays, and Fridays. Please attend your assigned session.

Main topics

• Brief review of linear algebra

• Real Fourier series expansion

• Inner product spaces of functions; Fourier series expansion as an expansion in an orthogonal basis

• Odd & even functions

• Complex Fourier series

• Fourier transform

• Dirac delta function

• Applications of Fourier analysis

• First order ordinary differential equations: linear, separable

• Second order ordinary differential equations: linear

• Reduction of order method

• Method of undetermined coefficients

• Variation of parameters

• Forced and damped harmonic oscillators

• Series solutions for ordinary differential equations

Summary sheets

Mathematical background

Fourier analysis (updated 31/10/17)

Differential equations

Reference books

No single text will be followed very closely. Many books and online resources cover this material well. The following are readily available in multiple copies in Hamilton Lending Library.

• Kreyszig, Advanced Engineering Mathematics (older editions are more complete for our topics)

• James, Advanced Modern Engineering Mathematics (for Fourier analysis)

• Croft & Davison: Mathematics for Engineers (more elementary than the others and not sufficient for all course material, but with step-by-step guided examples)

80% final exam, 20% continuous assessment.

Continuous assessment consists of problem sets given and completed in tutorial sessions, with collaboration among students and help from the TA.

Problem sheets with their solutions are posted here in the week following their presentation in tutorials.

Oct 4-6 Problems 1 Solutions 1

Oct 11-13 Problems 2 Solutions 2

Oct 18-20 Problems 3 Solutions 3

Oct 25-27 Problems 4 Solutions 4

Nov 1-3 Problems 5 Solutions 5

Nov 15-17 Problems 6 Solutions 6

Nov 22-24 Problems 7 Solutions 7

Nov 29-Dec 1 Problems 8 Solutions 8

Dec 6-8 Problems 9 Solutions 9

Past years' tutorial sheets and solutions are available here.

Topics covered in lectures

Sep 26: Introduction to the module. Preview of Fourier series. Kronecker delta.

Sep 28: Review of linear algebra: vector space, linear independence, basis, dimension, inner product.

Sep 29: An inner product on functions. Orthogonality, orthogonal projection.

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Oct 3: Orthogonality of Fourier modes, Euler formulas for Fourier coefficients from orthogonal projection, Dirichlet conditions for existence and convergence of Fourier series.

Oct 5: Discontinuity points and Gibbs phenomenon. Fourier series of sine wave and square wave.

Oct 6: Parity of functions. Application of parity to real Fourier series. Fourier series of the half-rectified sine wave.

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Oct 10: Half-range expansions and application to triangle waves. Complex inner product spaces. Orthogonality of complex Fourier modes.

Oct 12: Review of series and Kronecker delta. Euler formulas for complex Fourier coefficients. Equivalence to real Fourier series. Example of square wave.

Oct 13: Checks of the square wave series. Odd & even functions summed over integers. Statement and proof of Parseval's theorem.

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Oct 17: Parseval's theorem on the square wave, and applications. Fourier Transform definition, existence conditions, and frequency interpretation. Example of square pulse.

Oct 19: Fourier transform as continuum limit of complex Fourier series. Fourier inversion theorem. Applications: frequency filters, image processing. Interpretations of magnitude and phase of the Fourier transform.

Oct 20: Properties of the Fourier transform. Amplitude modulation. Example of exponential decay.

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Oct 24: Discrete Fourier Transform (DFT). Definition of Dirac delta.

Oct 26: Calculations with Dirac delta. Step function (Heaviside theta). Dirac delta in the Fourier transform.

Oct 27: Summary of Fourier analysis. Parseval's theorem for Fourier transform and DFT. Gaussian distribution and the Fourier transform.

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Oct 31: Classification of differential equations.

Nov 2: First order ODE: separable and linear homogeneous.

Nov 3: First order ODE: linear nonhomogeneous. Bernoulli differential equations.

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Nov 14: Second order linear homogeneous. Vector space of solutions. Reduction of order method.

Nov 16: Second order homogeneous with constant coefficients.

Nov 17: Damped harmonic oscillator.

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Nov 21: Euler-Cauchy equations. The Wronskian and Abel's Theorem.

Nov 23: Second order linear nonhomogeneous. Method of undetermined coefficients.

Nov 24: Forced damped harmonic oscillator.

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Nov 28: Method of variation of parameters.

Nov 30: Series solutions.

Dec 1: Airy equation. Existence and convergence of series solutions.

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Dec 5: Legendre equation, Legendre polynomials. Fuchs's theorem, Frobenius method.

Dec 7: Second solutions in Fuchs's theorem and the Frobenius method.

Dec 8: Example of second series solution about a regular singular point. Bessel equation and Bessel functions.

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Dec 12, 14, 15: Review

TCD School of Mathematics