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