| Course Description : | Open mapping theorem. Positive harmonic functions. The Phragmen-Lindelof method and interpolation. Approximation by rational functions: Runge’s theorem, Cauchy’s theorem, simply connected regions. Zeros of holomorphic functions: infinite products, the Weierstrass factorization theorem, the Mittag-Leffler theorem, Jensen’s formula, Blaschke products, the Muntz-Szasz theorem. Analytic continuation: Regular points and singular points, continuation along curve, The mondormy theorem, construction of a modular function, The Picard theorem. H^p-spaces(Hardy-spaces): Subharmonic functions, the space H^p and N ( Nevanlinna class), the theorem of F. and M. Riez, factorization theorems, the shift operator, conjugate functions. |