| Week 13 |
Review. |
| Week 12 |
The Fourier transform. Application to the heat equation on the line; Qualitative discussion. The convolution product and relation to the Fourier transform. The Poisson summation formula. |
| Week 11 |
Applications of the Argument Principle. Fourier series and application to the heat equation on a ring. |
| Week 10 |
Laurent series; Definitions, convergence in annuli and examples. Zeros, poles and essential singularities; The Argument Principle. |
| Week 9 |
More applications of the residue theorem. Series; Rational functions of trigonometric polynomials; Using branch cuts. |
| Week 8 |
Isolated singularities and the power series representation around poles. Meromorphic functions; Residues; The residue theorem; Computation of sums of series. |
| Week 7 |
The fundamental theorem of algebra. Power series. Analytic functions. Holomorphic functions are analytic and analytic functions are holomorphic. Zeros and poles. |
| Week 6 |
Cauchy's integral formula and first consequences. Holomorphic functions are infinitely differentiable; Cauchy estimates; The mean value property; The maximum modulus principle; Rouché's theorem. |
| Week 5 |
Contour integration. Examples and bounds; Consequences of the existence of an antiderivative; Cauchy's theorem and its proof; Consequences and examples. |
| Week 4 |
Trigonometric and hyperbolic functions, and equations involving them; Powers and roots. Dynamics of oscillating systems. Smooth paramatrised curves and contour integration; First examples. |
| Week 3 |
More on holomorphic functions and the Cauchy-Riemann eqations; Harmonic functions; The complex exponential and its properties; The logarithm: principal value and multivalued function, branch points and branch cuts. |
| Week 2 |
Complex functions. Functions as mapping of domains; Contunuity; Differentiability; Holomorphic functions; The Cauchy-Riemann equations; Examples. |
| Week 1 |
The complex numbers. Introduction and motivation; the complex plane; polar coordinates; modulus, conjugate and argument; the complex exponential; boundedness, connectedness, path connectedness. |