| Week |
Date |
Lectures |
Exercises |
(To hand in on Thursday in week n+1 if not specified) |
| 17 |
21-04-15 |
Complex numbers. Complex valued functions. Holomorphic
functions. The Cauchy-Riemann equations. Angles under holomorphic maps. |
I§1: 4, 5, 7, 8. I§2: 2, 8, 10, 13. Midterm 2012 Exercises 1 and 2 Exam 2010 Problem 1 |
|
| 17 |
23-04-15 |
Power series. Convergence
radius. Analytic functions. Differentiation of power series. Analytic ⇒ Holomorphic. |
I§2: 3, 4. I§3: 1, 2, 3,
4. I§4: 5. Midterm 2011 Exercise 5 |
I§2:
11,12 (p. 12) II§2: 8 (p. 59) |
| 18 |
28-04-15 |
Formulas for the
convergence radius. The set of zeroes of an analytic function is discrete. Analytic continuation. |
I§4: 6, 7. II§1: 1, 3,
4. II§2: 1, 3, 4, 5, 10. Retake 2013 Exercise 2 Midterm 2013 Exercise 5 |
|
| 18 |
30-04-15 |
Sums, products, and compositions of convergent power
series. Inverse function of an analytic function. |
II§4: 1, 2. II§2: 7, 11. II§3: 5. Prove the cases 0 and ∞ of Thm 2.6 (p. 55). Midterm 2012 Exercise 5 |
II§2:
6 (p. 59) Hand in |
| 19 |
07-05-15 |
Open mapping theorem. Maximum modulus principle. | II§5: 1-6 II§6: 1-6. Midterm 2013 Exercise 3 |
|
| 20 |
12-05-15 |
Connected topological
spaces. Integrals along piecewise-differentiable paths. Local primitive of a holomorphic function. |
III§2: 1, 3, 4, 8, 9,
10. Retake 2011 Exercise 3 |
II§3:
1 (p. 67) III§2: 7 (p. 103) |
| 21 |
19-05-15 |
Integrals along continuous
paths. Invariance of the integral under homotopies of paths and of loops. Global primitives of holomorphic functions. |
III§2: 2, 6.
III§5: 1-4. III§6: 1-4. Retake 2012 Exercise 2 |
|
| 21 |
21-05-15 |
Local Cauchy Integral Formula. Holomorphic ⇒ Analytic. | III§6: 5, 6, 7, 8. III§7:
1, 3. Midterm 2011 Exercise 4 |
III§2:
11 Hand in |
| 22 |
26-05-15 |
Laurent series. Residues. Isolated singularities: Removable singularities, poles, essential singularities. |
V§2: 4, 5, 8, 9, 11, 13. Endterm 2011 Bonus Exercise (a) |
|
| 22 |
28-05-15 |
Winding number. Homologous
paths. Chains. Global Cauchy Theorem. |
V§3: 1-5. Endterm 2011 Exercise 1 |
V§2: 14
(p. 165) V§3: 6 (p. 171) |
| 23 |
02-06-15 |
Calculation of residues.
Residue Formula, argument principle, and Rouche Theorem. |
VI§1: 15, 16, 18, 19, 26,
28. Endterm 2013 Exercise 3 |
|
| 23 |
04-06-15 |
Cauchy Integral Formula. Computation of improper definite integrals - I. |
IV§2: 2. VI§2: 1, 9, 10,
11. Retake 2011 Exercise 4 |
VI §1:
33 (p.189) Hand in |
| 24 |
09-06-15 |
Computation of improper
definite integrals - II. |
VI§2: 8, 14, 15, 18. Retake 2014 Exercise 3 |
|
| 24 |
11-06-15 |
Fractional linear
transformations. |
VII§5: 1-5, 11. Endterm 2013 Exercise 1 |
Hand
in VII §5: 12 (p.238) |
| 25 |
16-06-15 |
Analytic automorphisms of the unit disc and the
upper half-plane. Statement of the Riemann Mapping Theorem. Applications of complex analysis to Hydrodynamics. |
VI§1: 37, 38.
VII§2: 1, 2. VII§4: 2. Retake 2014 Exercise 4 |
|
| 25 |
18-06-15 |
The Riemann Zeta function and the Riemann Hypothesis. | Remaining problems from old
exams |
|
| 27 |
Exam: Tuesday 30-06-2015 ,
EDUC zaal BETA, 09.00 - 12.00 |
|||
| 30 |
Retake: Tuesday 21-07-2015, EDUC zaal BETA, 09.00 - 12.00 | |||