MATHEMATICAL METHODS OF PHYSICS WITH EXERCISES
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
MODULE I: 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.
MODULE II: Basic elements of linear algebra, and of the theory of finite dimensional vector spaces
MODULE I: The one assessment is the final written examination, which involves the solution of 4/5 questions/problems in 3/4 hours.
MODULE II: Written exam in which students must solve three exercises, each divided by two to four issues, of the same type as those performed in the classroom.
An oral test follows in which the student must demonstrate to have reached a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use, beyond the mere formal manipulation.
MODULE I: 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.
MODULE II: It is expected that the students of this course will acquire a certain operational familiarity with those mathematical tools which, although they have long been commonly used in quantum mechanics, are nevertheless advanced enough to exceed the limits of basic courses in mathematical analysis; nevertheless maintaining a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use of it, beyond the pure and simple formal manipulation.
MODULE I: 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)
MODULE II: The course consists of a synthetic but logically coherent introduction to the elements of Functional Analysis on which the mathematical formalism of quantum mechanics is based. The traditional elementary theory of Hilbert spaces will be accompanied by an "effective" introduction to the theory of temperate distributions. Without renouncing logical coherence, the use of abstract aspects of the theory of topological vector spaces will be minimized, while the emphasis will be placed on the most frequently used elementary distributional techniques, such as the Fourier transform, and Green functions.
MODULE I: 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)
MODULE II: General notion of functional space. Topological vector spaces. Normed and pre-Hilbertian spaces, Cauchy-Schwarz inequality, parallelogram identity, polarization identity. Banach and Hilbert spaces. Spaces of sequences. Synthetic introduction to the abstract measure theory and integration theory. Taking limit under the integral sign. Quadratic mean convergence. Spaces of square summable functions. Subspaces of a Hilbert Space; Theorem of projections. Orthogonal decomposition. Orthonormal systems and Hilbertian bases. Generalized Fourier series. Separable spaces. Hilbertian isomorphism; Continuous linear operators. Representation of continuous linear operators. Algebra of continuous operators: unitary operators, projectors, adjoint operators; convergence of series of operators. Fourier series for periodic functions. Trigonometric series. Fourier integral; elementary properties, the Riemann-Lebesgue lemma. Fourier transform in the Schwarz space of fast decreasing test functions. The Hermite basis and its basic properties. The Fourier-Plancherel transform. Tempered distributions as weak limit of square summable functions. Regular and singular distributions. The Dirac’s delta. Distributional derivatives. The Principal Part P1/x distribution. The pointlike charge potential. Further operations: change of variables, product, tensor product. The Fourier transform of tempered distributions; computational rules; explicit examples. Convolution of distributions. Fundamental solutions of a linear differential operator. Fundamental solutions, and the Cauchy problem; Green functions. Fundamental solutions for the diffusion equation, the Schroedinger equation for the free particle, and the wave equation.
MODULE I: 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.
MODULE II: Written notes provided by the teacher during the course
MODULE I: The course consists of 64lessons . Of these, approximately 50 will be devoted to core topics; the rest, to exercises, and additional topics.
MODULE II: Frontal lessons and exercises