MATHEMATICAL METHODS OF PHYSICS WITH EXERCISES MOD. I
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
Basic concepts in the theory of Metric Spaces, and in the Calculus for functions of one real variable; elementary notions about partial derivatives. No preparatory constraint.
The one assessment is the final written examination, which involves the solution of 4/5 questions/problems in 3/4 hours.
The course is meant to provide an introduction to Complex Analysis, or, more precisely, to the theory of functions of one complex variable. This is a fundamental subject, that naturally supplements the basic courses in Calculus, and is more or less relevant to all mathematicians, whatever their specialist inclination, and to physicists, whether applied or theoretical. Students will be led to realize that all basic functions of Calculus, originally introduced as functions of a real variable, are more naturally defined as functions of a complex variable; and that in this way a much deeper insight in their basic structure is gained. At the same time they will be introduced to some calculation tools based on complex analysis, which are of frequent use in applied mathematics and in physics. These include widely used techniques such as path integration, power series, and their use in the solution of ordinary differential equations.
Review of basic notions about the structure of the field of complex numbers. Definition of elementary trascendental function of one complex variable. Holomorphic functions; conditions of Cauchy and Rkiemann, conformal representations. Rules of complex differential calculus. Inverse function, roots, and logarithms. The extended complex plane, and the Riemann sphere. (10h)
Paths in a metric space; regular pahs in the complex plane. Path integrals. Conservative vector fields. Holomorphic functions, as vector fields; the Cauchy theorem. Integrals of the Cauchy type, and Cauchy's integral formula. Harmonic functions. Principle of maximum modulus, and Liouville's theorem.(10 h)
Power series. Analytic functions. Some noteworthy power series. (6h)
Isolated singular points; Laurent expansion; classification of isolated singularities. The Residue Theorem, and its applications. (6h)
The fundamental theorem on Analytic Continuation. Analytic continuation along a path ; monodromy, and polidromy; complete analytic functions, and Riemann surfaces. Integrals of polydromic functions. The Gamma function. (5h)
Linear Ordinary differential equations of 2nd order; existence and uniqueness of local solutions ; analytic continuation of local solutions ; structure of the space of solutions. Singular points: the Euler equation. Fundamental solutions. Regular singularities, and solutions in their neighborhood. The Bessel equation; Bessel functions of 1st and 2nd kind. Equations in the Fuchs class; the equation of Gauss, and the Hypergeometric function. (13h)
Review of basic notions about the structure of the field of complex numbers. Definition of elementary trascendental function of one complex variable. Holomorphic functions; conditions of Cauchy and Rkiemann, conformal representations. Rules of complex differential calculus. Inverse function, roots, and logarithms. The extended complex plane, and the Riemann sphere. (10h)
Paths in a metric space; regular pahs in the complex plane. Path integrals. Conservative vector fields. Holomorphic functions, as vector fields; the Cauchy theorem. Integrals of the Cauchy type, and Cauchy's integral formula. Harmonic functions. Principle of maximum modulus, and Liouville's theorem.(10 h)
Power series. Analytic functions. Some noteworthy power series. (6h)
Isolated singular points; Laurent expansion; classification of isolated singularities. The Residue Theorem, and its applications. (6h)
The fundamental theorem on Analytic Continuation. Analytic continuation along a path ; monodromy, and polidromy; complete analytic functions, and Riemann surfaces. Integrals of polydromic functions. The Gamma function. (5h)
Linear Ordinary differential equations of 2nd order; existence and uniqueness of local solutions ; analytic continuation of local solutions ; structure of the space of solutions. Singular points: the Euler equation. Fundamental solutions. Regular singularities, and solutions in their neighborhood. The Bessel equation; Bessel functions of 1st and 2nd kind. Equations in the Fuchs class; the equation of Gauss, and the Hypergeometric function. (13h)
Complete lecture notes written in LaTex by the teacher are available online. Suggested textbook for a deeper study:
John B. Conway, Functions of One Complex Variable.
The course consists of 64lessons . Of these, approximately 50 will be devoted to core topics; the rest, to exercises, and additional topics.