Math 3770H: Complex Analysis I

Fall 2008

Instructor: Nikolai Dokuchaev

Lecture notes, assignments, news, and announcements, will be being posted on WebCT.

Synopsis:

• Complex arithmetic and the complex plane; geometric interpretation.
• Complex functions: polynomials, the exponent, logarithm, inversion. Branch points. Applications for linear ordinary differential equations.
• Limits and series. Power series. Radius of convergence.
• Differentiation: Holomorphic functions. The Cauchy-Riemann Equations. Elliptic equations and holomorphic functions.
• Complex contour integrals; Cauchy's theorem.
• Taylor series. Analyticness of holomorphic functions. Zeros and identity theorem. Maximum principle.
• Laurent series. classification of singularities. The Cauchy Residue formula. Calculus of residues. Application for real integrals.
• Winding numbers, Argument Principle, Rouche's Theorem. Counting of roots using winding number. The Fundamental Theorem of Algebra.
• Fourier transform, Laplace transform, Fourier series, Z-transform. Frequency analysis for linear ordinary differential equations; Hardy spaces,
• Meetings: Three lectures and one tutorial weekly