Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 623: Complex Analysis

This is an introductory course in complex analysis, intended for advanced undergraduate and beginning graduate students.

Student Body:
Math, physics and eng majors. Graduate students in related areas
 
Background and Goals: 

This is an introduction to the theory of analytic functions of one complex variable. Attention is given to the techniques of complex analysis as well as the theory. It is highly recommended  for math majors and also suitable for students in the physical sciences and engineering.

Alternatives: 

N/A

Subsequent Courses: 

graduate courses

Course Content: 
  • The algebra of complex numbers; fractional powers.
  • Logarithm and power functions; exponential and trigonometric functions.
  • Analyticity; Cauchy-Riemann equations.
  • Integrals and Cauchy's Theorem and Formula
  • Morera's theorem; maximum modulus theorem; Liouville's theorem; Fundamental Theorem of Algebra.
  • Taylor series and Laurent series; regions of convergence, absolute and uniform convergence
  • The calculus of residues: isolated, removable, polar, and essential singularities; behavior of the function near an isolated singularity; calculating residues; evaluation of real integrals
  • Schwarz lemma and hyperbolic geometry
  • Harmonic functions: Laplacian; relation to analytic functions; conjugate harmonic functions; Dirichlet problem; Schwarz reflection principle; applications.
  • Conformal mappings: fractional linear transformations; the geometric nature of the power, exponential, and logarithmic maps; Riemann Mapping Theorem.
credits: 
3. (N-A)
semester: 
Fall
prereqs: 
Math 321 or 521 or Consent of Instructor

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, Wi  53706

(608) 263-3054

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