Week 13 |
The Fourier transform: some properties; Application to the solution of the heat equation; Review of the term's material. |
Week 12 |
From Taylor series to Fourier series; The Fourier transform. |
Week 11 |
Cauchy's integral formula in an annulus; Computing Laurent series: Examples; The Argument Principle; The winding number; Applications. |
Week 10 |
Summary and further applications of the Residue Theorem; Short review of the midterm; Laurent series and the classification of singularities. |
Week 9 |
The Residue Theorem; Computing residues; Examples; Summary; Application: evaluating convergent series. |
Week 8 |
Application of the maximum modulus principle to count zeros; The fundamental theorem of algebra; Taylor series; Holomorphic functions are analytic; Zeros, poles and their multiplicities; The residue. |
Week 7 |
Cauchy's integral formula (updated notes); Discussion and an example; Holomorphic functions are infinitely differentiable; The mean value theorem; The maximum modulus principle; The case of harmonic functions. |
Week 6 |
Cauchy's theorem (Notes and a proof to be found here); Examples; Review of the midterm. |
Week 5 |
Parametrized curves; Definition and examples; Integration of complex functions; Definition and examples; An upper bound; The case of functions with an antiderivative; Path independence. |
Week 4 |
Trigonometric and hyperbolic functions; The multivalued logarithm; The principal branch of general powers; The multivalued square root; Outlook on Riemann surfaces; Dynamics of driven oscillating systems. |
Week 3 |
The Cauchy-Riemann equations; Conjugate harmonic functions; Examples; The exponential and the logarithm. |
Week 2 |
More on the complex exponential; Subsets: bounded, open connected; Complex functions; Examples; Continuity: Definition and examples; Differentiability; Holomorphic functions. |
Week 1 |
Introduction; The set of complex numbers; The complex conjugate, inverses; The absolute value and the argument; The complex exponential; Trigonometric identities. |