# Part IB - Complex Methods

## Lectured by R. E. Hunt, Lent 2016

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 Analytic functions
- 1.1 The complex plane and the Riemann sphere
- 1.2 Complex differentiation
- 1.3 Harmonic functions
- 1.4 Multi-valued functions
- 1.5 Mobius map
- 1.6 Conformal maps
- 1.7 Solving Laplace's equation using conformal maps
- 2 Contour integration and Cauchy's theorem
- 2.1 Contour and integrals
- 2.2 Cauchy's theorem
- 2.3 Contour deformation
- 2.4 Cauchy's integral formula
- 3 Laurent series and singularities
- 4 The calculus of residues
- 4.1 The residue theorem
- 4.2 Applications of the residue theorem
- 4.3 Further applications of the residue theorem using rectangular contours
- 4.4 Jordan's lemma
- 5 Transform theory