## Fall 2013 - N. Reshetikhin

section: 001, course control number: 54282 office: 917 Evans Hall
email: reshetik at math berkeley edu
Classes: MWF, 2-3pm, 2 Evans Hall
Midterm 1: September 23
Midterm 2: October 30
Final: group 15, Th, Dec 19, 3-6pm
Office Hours: Monday 3:30-5:00pm

## Enrollment Questions

If you want to switch to another section, you have to do it yourself on TeleBears. If you have problems with this ask Thomas Brown "thomasbrown at berkeley dot edu"

## 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. Any edition for the last ten years will work.

## The syllabus

Here is the tentative list of topics in the order they will appear in the course. 1. Introduction. 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 derivatives. Partial derivatives. 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. Problem solving. 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. Problem solving. 25. "Multivalued" functions: Log, fractional powers of z etc. Analytical continuation. 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.

## Lecture notes and homework

Lecture notes can be found here here .
The list of homework and of solutions to past homework are here . Homework 1 should be turned in on Friday, September 6. After this homework should be turned in on Wednesdays.
The final grade will be computed as 15%hw + 25% M1 + 25% M2 + 35% Final.
Practice exams for midterms and the final with some solutions can be found here .