The calendar below provides information on the course’s lecture (L) and exam (E) sessions.
| SES # | TOPICS |
|---|---|
| L1 | The algebra of complex numbers: the geometry of the complex plane, the spherical representation |
| L2 | Exponential function and logarithm for a complex argument: the complex exponential and trigonometric functions, dealing with the complex logarithm |
| L3 | Analytic functions; rational functions: the role of the Cauchy-Riemann equations |
| L4 | Power series: complex power series, uniform convergence |
| E1 | First in-class test |
| L5 | Exponentials and trigonometric functions |
| L6 | Conformal maps; linear transformations: analytic functions and elementary geometric properties, conformality and scalar invariance |
| L7 | Linear transformations (cont.): cross ratio, symmetry, role of circles |
| L8 | Line integrals: path independence and its equivalence to the existence of a primitive |
| L9 | Cauchy-Goursat theorem |
| L10 | The special Cauchy formula and applications: removable singularities, the complex Taylor’s theorem with remainder |
| L11 | Isolated singularities |
| L12 | The local mapping; Schwarz’s lemma and non-Euclidean interpretation: topological features, the maximum modulus theorem |
| L13 | The general Cauchy theorem |
| L14 | The residue theorem and applications: calculation of residues, argument principle and Rouché’s theorem |
| L15 | Contour integration and applications: evaluation of definite integrals, careful handling of the logarithm |
| L16 | Harmonic functions: harmonic functions and holomorphic functions, Poisson’s formula, Schwarz’s theorem |
| E2 | Second in-class test |
| L17 | Mittag-Leffer’s theorem: Laurent series, partial fractions expansions |
| L18 | Infinite products: Weierstrass’ canonical products, the gamma function |
| L19 | Normal families: equiboundedness for holomorphic functions, Arzela’s theorem |
| L20 | The Riemann mapping theorem |
| L21-L22 | The prime number theorem: the history of the theorem and the proof, the details of the proof |
| L23 | The extension of the zeta function to C, the functional equation |
| E3 | Final exam |
