Analysis 2b

2022/2023
Programme:
Mathematics Education
Year:
2 year
Semester:
second
Kind:
mandatory
ECTS:
8
Language:
slovenian
Hours per week – 2. semester:
Lectures
4
Seminar
0
Tutorial
3
Lab
0
Prerequisites

Completed course Analysis 1.

Content (Syllabus outline)

Local parameterization. Arc length. First fundamental form of the surface. Surface area.
Scalar and vector fields. Gradient, curl, divergence and Laplacian operator. Curve and surface integrals. Gauss theorem. Stokes theorem and Green's formula. Application.
Holomorphic and harmonic functions. Cauchy-Riemann equations. Integrals of complex functions. Green's formula. Cauchy formula. Development in a power series. Uniqueness theorem. Cauchy estimates. Liouville theorem. Fundamental theorem of algebra. Open mapping theorem. Maximum principle. Laurent series expansion. Classification of isolated singular points. Meromorphic functions. Order of zero or pole. Residue theorem and applications. Argument principle. Rouché theorem. Holomorphic functions as maps. Conformal maps, elementary examples. Schwarz lemma. Holomorphic automorphisms of the disk and the plane. Riemann mapping theorem (without proof).
Fourier series.

Readings

Vidav: Višja Matematika II, DZS, Ljubljana, 1981.
T. M. Apostol: Calculus II : Multi-Variable Calculus and Linear Algebra with Applications, 2nd edition, John Wiley & Sons, New York, 1975.
J. E. Marsden, A. J. Tromba: Vector Calculus, 5th edition, Freeman, New York, 2004.
L. Ahlfors: Complex Analysis, 3rd edition, McGraw-Hill, New York, 1979.
J. B. Conway: Functions of One Complex Variable I, 2nd edition, Springer, New York-Berlin, 1995.
Suhadolc: Integralske transformacije/Integralske enačbe, DMFA-založništvo, Ljubljana, 1994.
A. Suhadolc: Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, DMFA-založništvo, 1998

Objectives and competences

Student becomes familiar with line and surface integrals, fundamentals of vector analysis, and the elementary theory of functions of one complex variable.

Intended learning outcomes

Knowledge and understanding: Undestanding of vector analysis, complex analysis and related topics. Application of methods in geometry and natural science.
Application: Analysis 2 is one of the fundamental courses in mathematical studies. It is a prerequisite for the courses Analysis 3, Complex analysis, Measure theory, Functional analysis, Probability and statistics, Analysis on manifolds.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to design the problem, select an appropriate method, solve the problem, and analyse the results on test cases. The ability to formulate a problem in mathematical language. Skills in using the domestic and foreign literature.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

ČERNE, Miran. Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces. American journal of mathematics, ISSN 0002-9327, 2004, vol. 126, no. 1, str. 65-87. [COBISS-SI-ID 12895577]
ČERNE, Miran, FORSTNERIČ, Franc. Embedding some bordered Riemann surfaces in the affine plane. Mathematical research letters, ISSN 1073-2780, 2002, vol. 9, no. 5-6, str. 683-696. [COBISS-SI-ID 12391257]
ČERNE, Miran. Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration. Arkiv för matematik, ISSN 0004-2080, 2002, vol. 40, no. 1, str. 27-45. [COBISS-SI-ID 11623513]
DRINOVEC-DRNOVŠEK, Barbara. Discs in Stein manifolds containing given discrete sets. Mathematische Zeitschrift, ISSN 0025-5874, 2002, vol. 239, no. 4, str. 683-702. [COBISS-SI-ID 11567449]
DRINOVEC-DRNOVŠEK, Barbara. Proper holomorphic discs avoiding closed convex sets. Mathematische Zeitschrift, ISSN 0025-5874, 2002, vol. 241, no. 3, str. 593-596. [COBISS-SI-ID 12076377]
DRINOVEC-DRNOVŠEK, Barbara. Proper discs in Stein manifolds avoiding complete pluripolar sets. Mathematical research letters, ISSN 1073-2780, 2004, vol. 11, no. 5-6, str. 575-581. [COBISS-SI-ID 13311065]
FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]
FORSTNERIČ, Franc. Noncritical holomorphic functions on Stein manifolds. Acta mathematica, ISSN 0001-5962, 2003, vol. 191, no. 2, str. 143-189. [COBISS-SI-ID 13138009]
FORSTNERIČ, Franc, ROSAY, Jean-Pierre. Approximation of biholomorphic mappings by automorphisms of C [sup] n. Inventiones Mathematicae, ISSN 0020-9910, 1993, let. 112, št. 2, str. 323-349. [COBISS-SI-ID 9452121]