COMPLEX ANALYSIS.

Catalog description

Notes: MATH 225 and MATH 460 are recommended. Pre-requisites: MATH 163 . The course proves relations between derivatives, integrals along curves, Maclaurin series, and singularities of complex-valued functions of a complex variable, in particular, theorems of Abel, Cauchy-Goursat, Green, Laurent, Liouville, Morera, Riemann and Rouch {\'{e}}. Applications include the solution of Laplace's partial differential equation by Green's functions (Cauchy's and Poisson's integral formulae) or Fourier Transforms. Detailed proofs of theorems also provide a theoretical foundations for the corresponding theorems from calculus with one or two variables: differentiation and integration of power series and Fourier series, differentiation relative to parameters of integrals along curves and the fundamental theorem of algebra.