ECTS - University of Pisa
| Modules | Area | Type | Hours | Teacher(s) | |
| METODI MATEMATICI 2 | FIS/02 | LEZIONI | 48 |
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The student who successfully completes the first part of the course will acquire a solid knowledge of the foundations of the theory of functions of a complex variable and of its applications to the evaluation of definite integrals and to two-dimensional potential problems in Physics. In the second part of the course, he/she will learn how to apply these techniques to study the properties of the so-called "Green function", a notion of fundamental importance in Physics with many applications in different fields. At the same time, he/she will acquire a basic knowledge of the foundations of the so-called "theory of distributions" (mainly of "tempered distributions" and their Fourier transforms), so putting on a mathematically rigorous basis the notion of the so-called "Dirac delta function" (plus other famous distributions).
The student will be assessed on his/her acquired knowledge from his/her ability to solve problems which are proposed in exercise classes and in a final written and oral exam.
The student will be able to solve exercises and problems (such as the evaluation of definite integrals and two-dimenstional potential problems in Physics) using the methods of the theory of analytic functions of a complex variable (including the so-called "conformal mapping"). Moreover, he/she will be able to solve exercises and problems in different fields (mainly, in electromagnetism) involving the so-called "Green functions", making also a proper use of the so-called "distributions".
The student will be assessed on his/her demonstrated skills from his/her ability to solve problems which are proposed in exercise classes and in a final written and oral exam.
The student will be able to undertake more advanced studies in many fields of Physics where the Mathematical Methods discussed in this course (i.e., the notions of analytic functions of a complex variable, of "Green functions" and of "distributions") are used.
The student will be assessed on his/her acquired maturity in managing the Mathematical Methods discussed in this course from his/her ability to solve problems which are proposed in exercise classes and in a final written and oral exam.
The student who attends this course should have already attended the course of "Mathematical Methods I", plus, of course, all the basic courses of Mathematics (given in the first two years) and also the course of "Physics II".
The student can be admitted to take the final examination only if he/she has already passed the final examination of the course of "Mathematical Methods I".
Delivery: face to face
Learning activities:
Attendance: Advised
Teaching methods:
A. FUNCTIONS OF A COMPLEX VARIABLE:
The notion of function of a complex variable: continuity, branch points and cuts.
Derivative of a function of a complex variable: definition and Cauchy-Riemann conditions.
Definition and properties of the so-called "analytic functions".
Integrals of a function with respect to a complex variable: fundamental properties and "Cauchy's theorems".
The formula of Cauchy's integral for an analytic function and some consequences. Integrals depending on a parameter and higher-order derivatives of an analytic function: "Morera's and Liouville's theorems".
Uniformlly convergent series of functions of a complex variable: general properties and "Weierstrass' theorem"
Power series of a complex variable and Taylor's series.
The zeros of an analytic function and the "uniqueness theorem". The analytic continuation of some elementary functions from the real axis to the complex plane. The notion of "Riemann surface".
The Laurent series and the classification of the isolated singular points of an analytic function.
The "residue" of an analytic function in an isolated singular point: the "fundamental theorem" of the residues' theory.
Evaluation of definite integrals by means of residues: "Jordan's lemma".
The relation between analytic and harmonic functions and the so-called "conformal mappings": some applications to two-dimensional potential problems in Physics.
B. GREEN FUNCTIONS AND ELEMENTS OF THE THEORY OF DISTRIBUTIONS:
Definition and properties of the so-called "Green function" (for time-independent linear systems): the spectral analysis and the "dispersion law". Properties of the Green function in the complex frequency plane for causal systems: the "Hilbert transformations" and the "Titchmarsh theorem". The physical example of the electric susceptibility of a dielectric material and the "Kramers-Kronig dispersion relations".
The general definition of a "distribution": "compact-support distributions" (E'), "tempered distributions" (S') and "Schwartz distributions" (D'). Various examples of distributions (e.g., the so-called "Dirac delta function").
The "weak convergence" of a sequence of distributions: examples of various sequences of distributions converging to the "Dirac delta function". Definition of the derivative and the Fourier transform of a tempered distribution.
The so-called "Principal Part distribution" [P(1/x)]. Products and convolutions of distributions.
Properites and examples of applications of distributions.
The Green function for the problem of the Coulomb potential produced by a given electric-charge distribution.
Examples of Green functions for problems with fixed boundary conditions.
The Green functions of the electromagnetic field and the derivation of the so-called "retarded potentials".
Recommended reading includes the following works:
1) G. Cicogna, "Metodi matematici della Fisica" (Second Edition, Ed. Springer, 2015): chapters 3, 4 (sections from 4.9 to 4.22) and 5 cover practically all the subjects of the course.
2) G. Cicogna, "Exercises and Problems in Mathematical Methods of Physics" (Second Edition, Ed. Springer, 2020).
3) A.G. Sveshnikov & A.N. Tikhonov, "The theory of functions of a complex variable" (Mir Publishers, 1982): it contains an in-depth and complete treatment of the subjects of the first part of the course.
Alternatively, one can also see:
4) V.I. Smirnov, "Corso di matematica superiore", vol. 3, parte II (Ed. Riuniti, 2011).
5) J.D. Jackson, "Elettrodinamica classica" (Ed. Zanichelli, 2001): it contains an in-depth treatment of all the physical examples discussed during the course.
Assessment methods:
The final written exam consists typically of two problems, one on the first part of the course and another on the second part of the course, which must be solved in about one-and-a-half/two hours. The student who passes the written proof (with at least a sufficient grade) is admitted to the final oral proof.
