Math 185 -- Complex analysis

sections: 001, 101, course control number: 58850, 58855 office: 917 Evans Hall
email: reshetik at math berkeley edu
Classes: 110 Barker Hall
Office Hours: Tuesday, Thursday 1-2:30 pm

The PDF file of the syllabus with the tantative list of topics for the whole course is here .

Brown, J. and Churchill, R. {\it Complex Analysis and Applications}, Eighth Edition. McGraw-Hill Science/Engineering/Math, 2008.

Tentative schedule

The schedule is evolving evolve as the course is developing. Below is the updated schedule. Corrections will be posted.

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. Square root . Lecture Notes . Homework which will be graded: p. 14 2, 7; p. 23 5; p. 44 2. Homework which will not be graded: p. 14: 14; p. 23: 11; p. 30: 6,7,8; p. 33: 1, 4, 6; p. 37: 1,2,3; p. 44: 3, 8. Solutions to the homework .

1. Analytic functions. Power series. 2. Power series. Problem solving session. 3. Problem solving session. 4. Paths in complex plane.Contour integrals. Lecture Notes. Homework . Solutions to the homework .

1. The existence of an antiderivative for an entire function. 2. Basic facts about entire functions. Problem solving. 3. Overview. Problem solving. 4. Midterm 1. Homework . Solutions to the homework . Lecture Notes . Practice midterm . Solutions to the midterm .

1. Cauchy theorem for domains. 2. Cauchy integral formula. Analytic functions are infinitely differentiable and all their derivatives are also analytic functions. 3. Liouville theorem. Taylor series. 4. Uniqueness theorem for analytic functions. Homework . Solutions to the homework . Lecture Notes .

1. Maximum modulus theorem. Morera theorem. 2. Isolated singularities. Laurent series. 3. Partial fraction decomposition. Problem solving. 4. Cauchy's residue theorem. Homework . Solutions to the homework . Lecture Notes . Practice problems for the Midterm 2 . Solutions to practice problems for the Midterm 2 . Homework .

1. Computations of integrals by residues. 2. Problem solving. 3. Review. 4. Midterm 2.

1. "Multivalued" functions: Log, fractional powers of z etc. 2. Integrals with branch cuts. Conformal mappings. Linear-fractional mappings. 3. Conformal mappings. Riemann mapping theorem. 4. Examples of conformal mappings. Analytic functions as vector fields. Homework . Lecture Notes . Practice Final .

1. Harmonic functions and the Dirichlet problem. 2. Review for the final. 3. Review for the final. 4. Final.