Prof. G. Dibitetto
(S1, compulsory, 8 ECTS)
Learning OutcomesSound understanding and ability in the use of mathematical methods that are the foundation to courses of modern physics and to current research in all branches of physics, both at the theoretical and experimental level.
Knowledge and Understanding
Understanding and mastery of advanced mathematical methods that are of use in other courses and in research contexts.
Applying Knowledge and UnderstandingAbility to prove theorems, derive mathematical properties, and make complex calculations. Moreover, the ability to identify the areas of applicability of the mathematical methods proposed in class, especially for the resolution of complex problems, even when concerning new topics. Comprehension of the course content, also regarding its more formal parts.
PrerequisitesKnowledge of mathematical analysis, geometry and calculus.
ProgramComplements of complex variable theory: Analytic and multivalued functions. Complex integrals. Pole expansion of meromorphic functions. Infinite product representation of complex functions. Local invertibility and the reciprocal of analytic functions. Asymptotic expansions: Integration by parts. Laplace method and Watson lemma. Stirling’s formula. Stokes phenomenon and analytic continuation. Stationary phase, steepest descent and saddle point techniques. Ordinary differential equations: Distribution theory. Green’s functions. Second order linear equations. Cauchy and Sturm-Liouville problems. Differential operators in Hilbert spaces. Equations in complex space. Power series method. Fourier and Laplace transforms: Discrete and integral transforms. Multidimensional cases. Special functions of physical science: Gamma, Diagamma, Polygamma, Beta and Zeta functions. Hypergeometric, confluent hypergeometric, Bessel functions. Legendre functions and spherical harmonics. Orthogonal polynomials. Partial differential equations: Classification, physical motivation and notable examples. Separation of variables and integral transform methods. Boundary value problems.
Description of how the course is conductedSee next point
Description of the didactic methodsThe course consists of theory lectures presented on blackboard. Every theory class is accompanied by exercises and examples, so that concepts do not remain too abstract and students can practice with actual calculations. Occasionally, slide presentations can be used, if there is a need to show data and plots, when illustrating examples of application of the studied mathematical methods to current research problems in physics.
Description of the evaluation methodsStudents need to pass a written intermediate test to access the final oral exam. The written test is intended to verify the students ability in applying the proposed mathematical methods and in performing calculations. The oral exam, on the other hand, is meant to verify the students’ ability in explaining clearly and correctly problems and information. It also provides a way to verify that the students have developed an adequate critical understanding of the subject, so that they can prove theorems, derive mathematical properties from first principles, and connect them to problems that are relevant in physics.
Adopted TextbooksM. Petrini, G. Pradisi, A. Zaffaroni, “A Guide to Mathematical Methods for Physicists: Advanced Topics”, World Scientific, 2018.
Recommended readings