MATHEMATICAL METHODS FOR PHYSICS: ELEMENTS OF COMPLEX ANALYSIS WITH EXERCISE SESSIONS
Knowledge of calculus in one and more real variables
The final exam will consist of a written exam in which 4-5 exercises must be solved in a time of 3 hours, and then an oral exam in which the learning level of the theory and the ability to connect the different conceptual aspects to the practical ones will be investigated and physical applications. The oral examination will follow the written one only if the mark in the latter is equal or larger than 18/30.
The main objective of the course is the introduction to complex analysis; more precisely, to the theory of functions of a complex variable. It is a fundamental topic, a natural supplement to the basic courses of Mathematical Analysis, and in varying degrees relevant for mathematicians of any orientation, as well as for physicists, both theoretical and applied. The course intends to guide the student to recognize that all the basic functions of Calculus, originally introduced as functions of real variables, are actually more naturally defined as functions of complex variables, and that this reveals their structure in a more deep than one can imagine remaining in the real field. At the same time the student will be trained to handle computational tools based on complex analysis, which are frequently used in applied mathematics and physics. He will be introduced to some widely applied techniques such as path integration, and the use of power series in solving ordinary differential equations.
Review of the basics of complex numbers. Definition of the elementary transcendent functions of a complex variable. Holomorphic functions: Cauchy-Riemann conditions, and conformal representations. Rules of derivation. Inverse functions, roots, and logarithms. The extended complex plane, and the Riemann sphere. Notion of path in a metric space. Regular walks in the complex plane. The path integral for functions of n real variables, and for functions of one complex variable. Conservative fields. Holomorphic functions, such as vector fields. Cauchy's theorem; Integrals of the Cauchy type, and the Cauchy integral formula. Harmonic functions. Principle of the highest modulus. Liouville's theorem. Power series; analytic functions. Notable series. Isolated singularities; Laurent expansion, classification of isolated singularities. Theorem of residues, and its applications. Fundamental Theorem on Analytical Prolongation; analytic extension along a path; monodromy, and polydromy; complete analytic functions, and Riemann surface. Integrals of polydrome functions. The Gamma function. Ordinary, linear, 2nd order differential equations. Local existence and uniqueness theorem; analytical extension of the solutions; structure of the solution space. Singular points. Euler's equation. Fundamental solutions. Regular singularities, and behaviour of the solutions in their neighbourhood. Bessel equation and functions. Equations of the Fuchs class; Gauss equation; Hypergeometric function; confluent hypergeometric equation.
The course consists of approximately 50 hours of frontal lessons, and approximately 14 hours of practice. In the latter, complementary topics will develop, and exercises and problems will be carried out.
Office hours: upon agreement with the teacher.
Professors
Borrowers
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Degree course in: MATHEMATICS