* Load is given in academic hour (1 academic hour = 45 minutes)
COURSE GOALS: Course goals are to acquire theoretical and practical knowledge in the complex analysis and in the theory of special functions.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
1. KNOWLEDGE AND UNDERSTANDING
1.1. demonstrate a thorough knowledge and understanding of the fundamental laws of classical and modern physics;
1.2. demonstrate a thorough knowledge and understanding of the most important physics theories (logical and mathematical structure, experimental support, described physical phenomena);
1.3. demonstrate knowledge and understanding of basic experimental methods, instruments and methods of experimental data processing in physics;
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1. identify and describe important aspects of a particular physical phenomenon or problem;
2.2. recognize and follow the logic of arguments, evaluate the adequacy of arguments and construct well supported arguments;
2.3. use mathematical methods to solve standard physics problems;
3. MAKING JUDGMENTS
3.1. develop a critical scientific attitude towards research in general, and in particular by learning to critically evaluate arguments, assumptions, abstract concepts and data;
4. COMMUNICATION SKILLS
4.1. communicate effectively with pupils and colleagues;
4.2. present complex ideas clearly and concisely;
4.4. use the written and oral English language communication skills that are essential for pursuing a career in physics and education;
5. LEARNING SKILLS
5.1. search for and use professional literature as well as any other sources of relevant information;
5.2. remain informed of new developments and methods in physics and education;
5.3. develop a personal sense of responsibility for their professional advancement and development;
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course on Mathematical Methods of Physics I, the student will be able to:
* define and analyse basic notions in the complex analysis (sequences, continuity, llimits, derivation, integrals and their properties, analytic functions, Taylor and Laurent series, Residue theory)
* define and analyse basic notion in the theory of functions of several variables (differential and partial derivatives)
* determine the Taylor and Laurent expansions of analytic functions
* calculate the integral of a complex function along a path
* calculate examples of real integrals using complex integral
* describe the properties of the Gamma and Beta functions and apply these functions in practical calculations
1) Complex numbers. Complex plane. Sequences of complex numbers
2) Complex functions. Continuity and limit
3) Functions of several variables. Partial derivates.
4) Derivation of a complex function. Analytic functions
5) Cauchy Riemann condition. Examples of analytic functions
6) Series of functions. Convergence of series of functions. Power series.
7) Complex integral
8) Cauchy theorem and Cauchy integral formula
9) Expansion of analytic function into Taylor series
10) Laurent expansion of analytic function
11) Isolated singularities. Classification of isolated singularities
12) Residue theorem. Application to real integrals
13) Gamma and Beta function
REQUIREMENTS FOR STUDENTS:
Students are expected to regularly attend lectures, exercises and do homework. Furthermore, students are required to pass two colloquiums during the semester, and to achieve at least 40% of the total number of points on them.
GRADING AND ASSESSING THE WORK OF STUDENTS:
Grading and assessing the work of students during the semesters:
* Two written exams
* Home works
Grading at the end of semester:
* final oral exam
Contributions to the final grade:
* 10% of the grade is carried by the results of the home works and on presence
* 60% of the grade is carried by the results of the two written exams
* the oral exam carries 30% of the grade.
- H. Kraljević, Matematičke metode fizike 1, Skripta, PMF-MO
- E. Freitag, R. Busam, Complex Analysis, Universitext, Springer, 2005.
- Š. Ungar, Kompleksna analiza, elektronicka skripta, http://web.math.hr/~ungar/kompleksna.pdf
- H. Kraljević, S. Kurepa, Matematička analiza IV, Tehnička knjiga, Zagreb, 1986.
- M R Spiegel, Schaum's Outline Series of Complex Variables, McGraw-Hill, 2009.
Mathematical Analysis 2