Unit MATHEMATICAL METHODS FOR PHYSICS
- Course
- Physics
- Study-unit Code
- GP005456
- Location
- PERUGIA
- Curriculum
- In all curricula
- Teacher
- Simone Pacetti
- Teachers
-
- Simone Pacetti
- Hours
- 84 ore - Simone Pacetti
- CFU
- 12
- Course Regulation
- Coorte 2021
- Offered
- 2022/23
- 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