The calendar below provides information on the course's lecture (L) and exam (E) sessions.
Course calendar.
| 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 |