Geometric interpretation of analytic properties of holomorphic functions. The Riemann sphere.
Schwarz and Schwarz Pick Lemmas. Poincare' metric and distance and associated geodesics..
Automorfphisms of the unitary disk. Automorphismsi of the complex plane and the Riemann sphere.
Linear fractional rasformations. Normal families.
Weierstrass Theorem. Hurwitz Theorem . Montel. Theorem. Uniformization (Riemann) Theorem.. Riemann surfaces and their classification.
L. V. Ahlfors, Complex Analysis, Third Edition, Mc Graw Hill 1979
J. B. Conway, Functions of One Complex Variabl, GTM Springer-Verlag, 1978
J. Milnor, Dynamics n One Complex Variable, Third Edition, Annals of Mathematics Studies, 2006
E. Vesentini, Capitoli scelti della teoria delle funzioni olomorfe. UMI 1980
Learning Objectives
Knowledge acquired:
Basic Knoweledge of the the theory of a complex variable.
Competence acquired:
Elements of complex function theory necessary for advanced topics in
Analysis, Geometry and Applied Mathematics
Skills acquired (at the end of the course):
Ability of using Complex Funcrion Theory in Analysis, Geometry and
Aplied Mathematics.
Prerequisites
Courses required: All the required courses of the Laurea In Mathematics
(first three year cicle)
Courses recommended: All basic courses courses in Algebra, Analisi and Geometria of the Laurea In Mathematics (first three year cicle)
Teaching Methods
Total hours of the course (including the time spent in attending lectures,
seminars, private study, examinations, etc...): 225
Hours reserved to private study and other indivual formative activities: 153
Hours for lectures: 72
Further information
Attendance of lectures, practice and lab:
Not mandatory
Teaching tools:
http://web.math.unifi.it/users/patrizio/DidaI/
Office hours:
Monday 14:30 and by appointment
Contact:
Viale Morgagni, 67/a - 50134 Firenze
Phone: 055 4237109
Fax: 055 4237165
E-mail: giorgio.patrizio@math.unifi.it
Web: http://web.math.unifi.it/users/patrizio/
Type of Assessment
Written and oral exam
Course program
Complex numbers and topology of C. Holomorphic functions and conformality. Power series and elementary functions. Integration along curves, Cauchy-Goursat Theorem, Cauchy formula and conseguences. Developements in power series, zeros of holomorphic functions, analytic continuation. Cauchy inequalities. Sequences of holomorphic functions. Liouville Theorem, Open mapping theorem, Maximum modulus theorem. Schwartz’s Lemma. Global Cauchy’s formulas, homotopy. Laurent series and isolated singularities. Riemann sphere and meromorphic functions. Residue, Argument principle, Rouché’s theorem, Hurwitz’ theorem. Conformal representation and Riemann mapping theorem. Applications of Differential Geometry to function theory. Elements of harmonic functions theory. Introductive questions of holomorphic dynamics.