Fall 2011 - N. Reshetikhin
section: 002,
course control number: 54407
office: 917 Evans Hall
email: reshetik at math berkeley edu
Classes: MWF, 3-4pm, 6 Evans Hall
Midterm 1: Sept. 23, 3-4pm, 2 Evans Hall
Midterm 2: Oct. 26, 3-4pm, 2 Evans Hall
Final: Dec. 13, 7-10pm, Final Exam Group 8
Office Hours: Tuesday, 2-3pm, 917 Evans Hall
email: reshetik at math berkeley edu
Classes: MWF, 3-4pm, 6 Evans Hall
Midterm 1: Sept. 23, 3-4pm, 2 Evans Hall
Midterm 2: Oct. 26, 3-4pm, 2 Evans Hall
Final: Dec. 13, 7-10pm, Final Exam Group 8
Office Hours: Tuesday, 2-3pm, 917 Evans Hall
Enrollment Questions
If you want to switch to another section, you have to do it yourself
on TeleBears.
Incompletes
Official University policy states that an incomplete can be
given only for valid medical excuses with a doctor's certificate
and only if, at the point the grade is given, the student has a
passing grade (a C or better). If you are behind in the course, an
incomplete is not an option!
Textbook:
Brown, J. and Churchill, R. {\it Complex Variables and Applications}, Eighth Edition. McGraw-Hill Science/Engineering/Math, 2008.
The syllabus
Here is the tentative list of topics in the order they
will appear in the course.
1. Introduction. Real and Complex numbers
(addition, multiplication, division, etc.). The complex
plane (complex numbers as vectors, polar coordinates.
2. A neighborhood of a point. Interior points, boundary points, boundary. Open sets, closed sets, closures. Connectedness (domain is connected). Bounded, unbounded domains. Accumulation points. Sequences of points and their limits. Closure is the addition of limit points to the set.
3. Complex values functions on a complex plain f(x,y)=u(x,y)+iv(x,y). Functions as mappings.
Continuous mappings. Inverse mappings can be `multivalued'.
Examples: polynomials in z and conjugate z, rational functions,
power series in z (analytic functions).
4. Directional derivative. Partial derivative.
Derivative in z. Cauchy-Riemann equation.
Holomorphic functions.
5. Analytic functions. Power series.
6. Power series. Problem solving session.
7. Problem solving session.
8. Paths in complex plane.Contour integrals.
9. The existence of an antiderivative for an entire function.
10. Basic facts about entire functions. Problem solving.
11. Overview. Problem solving.
12. Midterm 1.
13. Cauchy theorem for domains.
14. Cauchy integral formula.
Analytic functions are infinitely differentiable and all
their derivatives are also analytic functions.
15. Liouville theorem. Taylor series.
16. Uniqueness theorem for analytic functions.
17. Maximum modulus theorem. Morera theorem.
18. Isolated singularities. Laurent series.
19. Partial fraction decomposition. Problem solving.
20. Cauchy's residue theorem.
21. Computations of integrals by residues.
22. Problem solving.
23. Review.
24. Midterm 2.
25. "Multivalued" functions: Log, fractional powers of z etc.
26. Integrals with branch cuts. Conformal mappings. Linear-fractional mappings.
27. Conformal mappings. Riemann mapping theorem.
28. Examples of conformal mappings. Analytic functions as
vector fields.
29. Harmonic functions and the Dirichlet problem.
30. Review for the final.