Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Supplementary compulsory subjects
First Semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (64 hours)

Basic concepts in the theory of Metric Spaces, and in the Calculus for functions of one real variable; elementary notions about partial derivatives.

Basic elements of linear algebra and the theory of vector spaces in finite dimension.

Final Examination: 

The one assessment is the final written examination, which involves the solution of 4/5 questions/problems in 3/4 hours.

The final assessment will consist in a written examination, involving the solution of 4/5 problems in 3/4 hours, followed by a short colloquium.

Voto Finale

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, wathever their specialistic 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.

The course consists in an introduction to those elements of Functional Analysis on which the mathematical formalism of Quantum Mechanics is based. Thus, an elementary theory of Hilbert spaces will be presented, together with an "effective" introduction to the theory of tempered distributions.
It is expected that the students of this course will acquire a certain degree of familiarity with the mathematical tools commonly used in Quantum Mechanics, and an understanding underlying mathematical theory sufficient to ensure its critical use, going beyond pure and simple formal manipulations.

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. 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. Power series. Analytic functions. Some noteworthy power series. Isolated singular points; Laurent expansion; classification of isolated singularities. The Residue Theorem, and its applications. 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. 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, the Hypergeometric function, and the confluent hypergeometric equation.

General notion of functional space. Topological vector spaces. Normed spaces and prehilbertian spaces, Cauchy-Schwarz inequality, the parallelogram identity, the polarization identity. Banach and Hilbert spaces. Spaces of sequences. Synthetic introduction to the abstract theory of measure and integration. Taking limits under integral sign. Quadratic mean convergence. Spaces of square summable functions. Subspaces of a Hilbert space; the projection theorem. Orthogonal decomposition. Orthonormal systems Hilbertian bases. Generalized Fourier series. Separable spaces. Hilbertian isomorphism; linear and continuous operators. Representation of linear and continuous functionals. Algebra of continuous operators: unitary operators, projectors, adjoint operator; convergence of sequences of operators. Fourier series for periodic functions; rate of convergence. Trigonometric series. Fourier integral; elementary properties, the Riemann-Lebesgue lemma. Fourier transform in the Schwarz space of fast decrease test functions. The Hermite basis and its basic properties. The Fourier-Plancharel transform. The Hermite basis and its basic properties. The Fourier-Plancherel transform. Tempered distributions as weak limits of square summable functions. Regular and singular distributions. The Dirac delta distribution. Distributional derivatives. The Principal Part distribution P1/x. Potential of a pointlike charged particle. Further operations: change of variables, product, tensor product. Fourier transform of tempered distributions; rules for calculations; explicit examples. Convolution of distributions. Hilbert transform. 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.

Complete lecture notes written in LaTex by the teacher are available online. Suggested textbook for more detailed study:
John B. Conway, Functions of One Complex Variable.

Complete lecture notes written in LaTex by prof. Guarneri are available online. Suggested textbook for more detailed study:
Lokenath Debnath and Piotr Mikusinski, Hilbert Spaces with Applications.

The course, given by professor I. Guarneri, consists of 64 lessons. Of these, approximately 50 will be devoted to core topics; the rest, to exercises, and additional topics.

The course, given by professor S.L. Cacciatori, consists of 64 lessons. Of these, approximately 50 will be devoted to core topics; the rest, to exercises, and additional topics. Attending the course is strongly encouraged.

Office hours: to be agreed with the teachers.

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