Advanced mathematical analysis techniques link to modern developments in theoretical physics.
List of topics: Elements of complex analysis; Residue theorem and applications; Simple asymptotic techniques; Special functions; Linear differential equations on C; Green functions; Distributions (outline).
G. Pradisi, Lezioni di metodi matematici della fisica, Collana "Appunti", Edizioni della Normale, Pisa (2012)
L.V. Ahlfors, Complex Analysis, Mc. Graw-Hill, New York (1996)
E.T. Whittaker, G.N. Watson, A course of modern analysis, Cambridge University Press (1927, reissued 1996)
Notes on "Analisi funzionale" by Prof. D. Dominici, can be found at the link: http://theory.fi.infn.it/colomo/metodi
Additional textbooks will be suggested according to the different subjects to be treated.
Learning Objectives
Acquired knowledge: Advanced notions of complex variable analysis, differential equations and distribution theory.
Skills acquired (at the end of the course): The student must assimilate basic concepts of the course and master the involved techniques; The student must be able to solve analytic and numerical exercises related to the topics of the course.
Competence acquired: Possibility of understanding the basics of modern physical theories, both in their foundational and technical aspects.
Prerequisites
The course requires a good knowledge of mathematical analysis. It is strongly suggested having already attended the course of "Mathematical Methods for Physics", and, possibly, having given the exam.
Teaching Methods
6 CFU
Lectures hours: 48
The course contains lessons with a discussion of the theory subjects and resolution of exercises.
Further information
The first part of the course (3 CFU) is taught by F. Colomo, the second one (3 CFU) by G. Panico.
Office hours:
on demand,
colomo@fi.infn.it
giuliano.panico@unifi.it
Website: http://theory.fi.infn.it/colomo
Type of Assessment
Written exam with exercises on the topics presented during the course.
Course program
Elements of complex analysis: Functions of a complex variable; Polydrome functions; Cuts; Riemann surface; Analytical and holomorphic functions; Meromorphic functions; Integration on a path; Cauchy theorem; Taylor and Laurent expansion; Uniqueness theorem; Analytical continuation with examples; Borel transform; Conformal transformations; Moebius transformations; Riemann theorem and conformal transformations; 2D electrostatic problems in regions with non-trivial shape.
Residue theorem and applications: Residue theorem; Jordan lemma; Principal value; Residue at infinity; Applications; Logarithmic indicator theorem; Pole expansion; Expansion of Mittag-Leffler; Sommerfeld-Watson transform; Rouché's theorem; Infinite Weierstrass products and sums; Applications.
Special functions: Euler Gamma Functions; Representation of Hankel; Watson's lemma; Riemann Zeta function; Reflection relation.
Linear differential equations on C: Regular points; Solution by series; Regular singular points (or of Fuchs); Monodromy matrix; Point at infinity; Equations with 1 or 2 Fuchsian singularities; Euler equation; Equations with 3 Fuchsian singularities; Symbol of Papperitz-Riemann; Hypergeometric equation; Hypergeometric series; Hypergeometric polynomials; Confluence; Confluent hypergeometric.
Green functions: Wronskian method and Green function method; Green theorem; Boundary problems in one dimension; Applications; Electrostatics in 2D and its Green function.
Distributions (basic concepts): Metric / normed / Banach / Hilbert spaces; Linear functionals; Space D of test functions; Distributions on D; Operations on distributions; Tempered distributions; Sequences of distributions; delta-families; Principal value and identity of Cauchy; Fourier transform of
tempered distributions.