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 |