To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.
Curriculum
teacher profile teaching materials
Programme
Complex numbers; holomorphic functions and the Cauchy_Riemann formula. Some examples of holomorphic functions; the Riemann sphere and e the point at infinity. The linear fractional transformations. Integral of a complex function along a curve; index of a point with respect to a curve. Cauchy theorem; Cauchy formula. The Liouville theorem. The mean value thorem, the maximum principle and the principle of identity of holomorphic functions. The almost-uniform limit of holomorphic functions is holomorphic. Shcwarz lemma and the automorphisms of the disc. The metric of Poincare' on the disc and its geodesics. Laurent series; the general form of Cauchy theorem. Removable singularities; poles and essential singularities; the Casorati-Weierstrass theorem. Euler's product for the sine. Meromorphic functions. The argument principle and the theorem of Rouche'. Holomorphic maps are open; the almost uniform limit of univalent functions is eithe univalent or constant; Lagrange inversion formula. Harmonic functions; the mean value property, the maximum principle and Dirichlet problem; Poisson kernel; continuous functions with the mean value property are harmonic. Schwarz reflection principle. Analytic extension. Jensen's formula for the zeroes of a holomorphic function. Normal families and compactness for the almost-uniform topology. The Riemann mapping theorem. When two rings are conformally equivalent. The small theorem of Picard. Holomorphic functions and fluidodynamics.Core Documentation
W. Rudin, Real and complex Analysis, McGraw-Hill.Type of evaluation
Two mini-tests during the course; a written examination at the end for those who fail the mini-examination. Oral exam for everybody. teacher profile teaching materials
Mutuazione: 20410593 AC310-ANALISI COMPLESSA in Matematica L-35 BESSI UGO
Programme
Complex numbers; holomorphic functions and the Cauchy_Riemann formula. Some examples of holomorphic functions; the Riemann sphere and e the point at infinity. The linear fractional transformations. Integral of a complex function along a curve; index of a point with respect to a curve. Cauchy theorem; Cauchy formula. The Liouville theorem. The mean value thorem, the maximum principle and the principle of identity of holomorphic functions. The almost-uniform limit of holomorphic functions is holomorphic. Shcwarz lemma and the automorphisms of the disc. The metric of Poincare' on the disc and its geodesics. Laurent series; the general form of Cauchy theorem. Removable singularities; poles and essential singularities; the Casorati-Weierstrass theorem. Euler's product for the sine. Meromorphic functions. The argument principle and the theorem of Rouche'. Holomorphic maps are open; the almost uniform limit of univalent functions is eithe univalent or constant; Lagrange inversion formula. Harmonic functions; the mean value property, the maximum principle and Dirichlet problem; Poisson kernel; continuous functions with the mean value property are harmonic. Schwarz reflection principle. Analytic extension. Jensen's formula for the zeroes of a holomorphic function. Normal families and compactness for the almost-uniform topology. The Riemann mapping theorem. When two rings are conformally equivalent. The small theorem of Picard. Holomorphic functions and fluidodynamics.Core Documentation
W. Rudin, Real and complex Analysis, McGraw-Hill.Type of evaluation
Two mini-tests during the course; a written examination at the end for those who fail the mini-examination. Oral exam for everybody.