**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**

**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)

**Grade**

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

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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