Complex Variables I

Course Description

Complex differentiability, Cauchy-Riemann equations, power series and elementary functions. Cauchy's theorem, the Cauchy integral formula, Cauchy's estimates, Morera's theorem. Entire functions, Liouville's theorem. Convergence, differentiation, and integration of sequences and series of holomorphic functions. Local mapping properties of holomorphic functions: isolation of zeros, conformality, inverse function theorem, critical points. Elementary Riemann surfaces. Classification of singularities. Laurent series. The residue theorem and applications: meromorphic functions, the Mittag-Leffler theorem. Holomorphic functions on the Riemann sphere, Möbius transformations. Maximum modulus principle, Schwarz's lemma, conformal maps of the unit disc. The Riemann mapping theorem, the Schwarz reflection principle. Harmonic functions, harmonic conjugates. The Dirichlet problem and the Poisson kernel for the unit disc. Prerequisite: 26:645:611.

Code

26:645:621

Credits

3

Subject

Mathematical Sciences