 Calendar

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