# Part IB - Methods

## Lectured by D. B. Skinner, Michaelmas 2015

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Vector spaces
- 2 Fourier series
- 3 Sturm-Liouville Theory
- 4 Partial differential equations
- 4.1 Laplace's equation
- 4.2 Laplace's equation in the unit disk in R2
- 4.3 Separation of variables
- 4.4 Laplace's equation in spherical polar coordinates
- 4.5 Multipole expansions for Laplace's equation
- 4.6 Laplace's equation in cylindrical coordinates
- 4.7 The heat equation
- 4.8 The wave equation
- 5 Distributions
- 6 Fourier transforms
- 6.1 The Fourier transform
- 6.2 The Fourier inversion theorem
- 6.3 Parseval's theorem for Fourier transforms
- 6.4 A word of warning
- 6.5 Fourier transformation of distributions
- 6.6 Linear systems and response functions
- 6.7 General form of transfer function
- 6.8 The discrete Fourier transform
- 6.9 The fast Fourier transform*
- 7 More partial differential equations