Math 623 - Complex Analysis

• Prerequisites: Math 321 or 521.
• Frequency: Fall (I)
• Student Body: Math, physics and eng majors. Graduate students in related areas
• Credits: 3. (N-A)
• Recent Texts:
• Course Coordinator: Daniel Rider
• 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 particularly suitable for students in the physical sciences and engineering as well as math majors.
• Alternatives: N/A
• Subsequent Courses: graduate courses

Content coverage:

• 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
• Conformal mappings: fractional linear transformations; the geometric nature of the power, exponential, and logarithmic maps; Riemann Mapping Theorem.
• Harmonic functions: Laplacian; relation to analytic functions; conjugate harmonic functions; Dirichlet problem; applications.