Complex Analysis I

• Text: Real and Complex Analysis
• Author:Walter Rudin
• ISBN: 0070542341
• Publisher: McGraw-Hill

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Topics covered

Starting at Chapter 10 of the text.

1. Complex numbers
   • Basic properties
   • The Riemann Sphere
2. Differentiability
   • Holomorphic functions
   • Integration over paths
   • The local Cauchy theorem
   • The power series representation
   • The open mapping theorem
   • The global Cauchy theorem
   • The Calculus of Residues
3. Harmonic functions
   • The Cauchy-Riemann equations
   • The Poisson integral
   • The Mean-Value property
   • Boundary behavior of Poisson integrals
   • Representation theorems
4. The Maximum-Modulus Principle
   • The Schwartz Lemma
   • The Phragmen-Lindelöf method
   • An interpolation theorem
   • Boundary behavior of Poisson integrals
   • A converse of the Maximum-modulus principle
5. Approximation by Rational Functions
   • Runge's Theorem
   • The Mittag-Leffler theorem
   • An interpolation theorem
   • Simply-connected regions
6. Conformal Mapping
   • Preservation of angles
   • Linear fractional transformations
   • Normal families
   • The Riemann Mapping Theorem

Links

1. Complex Numbers home page
2. Complex Analysis Page
3. Graphics for complex analysis

Grading

Grades will be based on graded homework assignments given each week (mostly taken from the textbook).

Lecture Notes

Lecture notes in PDF format