Unit MATHEMATICAL METHODS FOR PHYSICS
- Course
- Physics
- Study-unit Code
- GP005456
- Curriculum
- In all curricula
- Teacher
- Simone Pacetti
- Teachers
-
- Simone Pacetti
- Hours
- 94 ore - Simone Pacetti
- CFU
- 12
- Course Regulation
- Coorte 2024
- Offered
- 2025/26
- Learning activities
- Caratterizzante
- Area
- Teorico e dei fondamenti della fisica
- Academic discipline
- FIS/02
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Analytic complex functions of complex variable Theorems for the contour integration in the complex plane Integral and series representations Fourier transforms Linear vector spaces Linear operators: definitions, representations and algebra Integral and differential equations
- Reference texts
- "Complex Analysis" S. Lang Springer Verlag "Complex Analysis" L.V. Ahlfors McGraw Hill "Metodi Matematici per la Fisica" C. Rossetti Levrotto e Bella editore "Introduction to Hilbert Spaces with Applications" L. Debnath and P. Mikusinski Academic Press
- Educational objectives
- Skill in handling complex analytic functions, i.e.: identification of singularities, asymptotic behavior, integral and series representations, as well as complex contour integration using basic theorems and lemmas. Ability in computing and using the Fourier transforms. Knowledge of the linear operator algebra in Hilbert spaces, giving special attention to Hermitian and unitary operators. Mastery in classifying integral and differential equations, in proving the existence and uniqueness of the solution and in using procedures to compute such solutions.
- Prerequisites
- Limits of functions. Differential and integral calculus. Numerical sequences and series.
- Teaching methods
- Frontal lessons and practical training.
- Learning verification modality
- Written and oral exam.
- Extended program
- Complex numbers: properties and applications to Physics Analytic functions Conformal mapping Zeros and singularities Integration of complex-valued functions Cauchy's theorem and integral formula Integration of infinite and infinitesimal arcs. Jordan's lemma Cauchy principal value and Sokhotsky-Plemelj formula The residue theorem Integral representation and series Convergence theorems Taylor and Laurent series The Mittag-Leffler theorem Analytic continuation Dispersion relations Infinite products The Euler's gamma function The Riemann's Zeta function Linear vector spaces The Schwarz inequality Banach and Hilbert spaces and vector series Linear operators and basis Hermitian and unitary operators Projection operators Eigenvectors and eigenvalues Representation of an operator and its adjoint Orthonormal bases and unitary transformations The eigenvalue equation and the diagonalization procedure Diagonalizable and normal operators Quantum mechanical observables Pauli matrices and their algebra The Lebesgue measure and integral Fourier series Quadratically integrable functions Convergence theorems for function sequences Generalized functions and the Dirac delta Fourier transforms Solving differential equations using Fourier transforms The Green function Integral equations Classical orthogonal polynomials