Mathematics IV

Enrollment into the second academic year.

The prerequisite for the theoretical exam is a positive result of the written exam.

Homework may also be one of obligations.

Holomorphic functions: extended complex plane, domains and paths, the Cauchy-Riemann equations, contour integrals, winding number, Cauchy’s formula, Liouvlle’s theorem, Laurent’s and power series, zeros of holomorphic functions, isolated singular points, logarithm and powers, the residue theorem, gamma function, open maps and the maximum modulus principle, Moebius transformations.

Harmonic functions: Poisson’s formula for the disk, harmonic functions in Rˆ3 (Dirichlet’s problem and Green’s function).

Fourier transform: convolution, basic properties , the inverse transform, Plancherel’s theorem, applications (to heat equation).

Power series solutions of differential equations of second order: ordinary and regular singular points, Bessel’s equation and functions, Legendre's equation, polynomials and associated functions, other orthogonal polynomials (Hermite and Laguerre).

Boundary value problems:the vibrating string (Fourier’s method of separation of variables, d’Alembert’s formula), the regular Sturm-Liouville problem, vibration of circular membrane, the Laplace equation on the ball.

G. B. Folland, Fourier Analysis and its Applications, AMS, Providence RI, 1992.

S. Lang, Complex Analysis, Springer, New York, 2003.

B. Magajna, Uvod v diferencialne enačbe, kompleksno in Fourierovo analizo, DMFA-založništvo, Ljubljana 2018.

G. F. Simmons, Differential Equations with Appl. And Historical Notes, Chapman and Hall/CRC, Boca Raton, 2016.

A. Suhadolc, Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, Matematični rokopisi 23, DMFA, Ljubljana, 1998.

E. Zakrajšek, Analiza III, Matematični rokopisi 21, DMFA-založništvo, Ljubljana 2002.

M. Braun, Differential Equations and Their Applications, 4th ed. Applied mathematical sciences 15, Springer-Verlag, New York 1993.

Students learn some tools of mathematical analysis that are important for applications in physics.

Knowledge and understanding:
We expect that students know important definitions and theorems, understand (and ideally be able to replicate) at least the easier proofs, and be able to apply this knowledge , e.g. in mathematical physics

Application:
The material of this course is indispensable for Mechanics, Mathematical physics and other courses in the program.

Reflection:
Students master some advanced topics in Mathematical Analysis and are able to apply them in physics.

Transferable skills:
Students learn to understand the usefulness of the abstract approach, are able to connect the acquired knowledge with what they already mastered. They also learn to use other written sources and the internet. They are able to identify and solve problems, hand in homework on time, and memorize the important topics.

Lectures, tutorials, homework (optional).

Exercise-based exam / 2 midterm exams instead of written exam, written exam, homework optional).
Oral exam or theoretical test
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Prof. Dr. Miran Černe:
1. M. Černe, M. Zajec, Boundary differential relations for holomorphic functions on the disc, Proc. Am. Math. Soc. 139 (2011), 473-484.
2. M. Černe, M. Flores, Generalized Ahlfors functions, Trans. Am. Math. Soc. 359 (2007), 671-686.
3. M. Černe, M. Flores, Quasilinear -equation on bordered Riemann surfaces, Math. Ann. 335 (2006), 379-403.

prof. dr. Bojan Magajna:
1. B. Magajna, C*-convex sets and completely positive maps, Integr. Equ. Oper. Theory 85 (2016) 37—62.
2. B. Magajna, Fixed points of normal completely positive maps on B(H), J. Math. Anal. Appl. 389 (2012) 1291--1302.
3. B. Magajna, The Haagerup norm on the tensor product of operator modules, J. Funct. Anal 129 (1995) 325—348.

prof. dr. J. Mrčun:
1. I. Moerdijk, J. Mrčun, On the developability of Lie subalgebroid, Adv. Math. 210 (2007), 1--21.
2. J. Mrčun, On isomorphisms of algebras of smooth functions, Proc. Amer. Math. Soc. 133 (2005), 3109--3113.
3. I. Moerdijk, J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math. 124 (2002), 567--593.