COURSE GOALS:
- acquire knowledge and understanding of the complex analysis and ordinary differential equations
- acquire operational knowledge from methods used to solve complex integrals and ordinary differential equations
- understand the usage of these mathematical methods in physics
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.2 evaluate clearly the orders of magnitude in situations which are physically different, but show analogies, thus allowing the use of known solutions in new problems;
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
4. COMMUNICATION SKILLS
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
5.2 remain informed of new developments and methods and provide professional advice on their possible range and applications
5.3 carry out research by undertaking a PhD
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course on Mathematical methods of physics 1, the student will be able to:
1. Find the Laurent series of a complex function
2. Solve the integrals of complex function with various methods (via different choices of contours)
3. Solve the real integrals using integration of complex functions
4. Compute series using integration of complex functions
5. Use gamma function in practice
6. Solve the linear ordinary differential equations of first order
7. Solve the linear ordinary differential equations of second order (method of variation of parameters, Frobenius' method)
8. Solve the linear ordinary differential equations of higher order with constant coefficients
9. Solve basic systems of linear ordinary equations
COURSE DESCRIPTION:
The Fall semester (each lesson 45 minutes)
1. Introduction: sets, functions of real variable [3 lessons]
2. Basic algebra with complex numbers, functions of complex variable, complex plane, Riemann surfaces [5 lessons]
3. Sequences and series of complex numbers [4 lessons]
4. Continuity and derivability of complex functions [3 lessons]
5. Integration of complex functions (Cauchy-Goursat theorem, Cauchy integral formula) [5 lessons]
6. Laurent series, singularities and residues [4 lessons]
7. Gamma functions [3 lessons]
8. Asymptotic expansions [3 lessons]
9. Linear ordinary differential equations of first order (methods of solution, uniqueness of solution) [4 lessons]
10. Linear ordinary differential equations of higher order (characteristic equations, Wronskian) [4 lessons]
11. Frobenius method [3 lessons]
12. Hypergeometric differential equation [2 lessons]
13. Systems of linear ordinary differential equations [2 lessons]
Exercises follow lectures by content:
The Fall semester (15 weeks)
1st week: set of complex numbers, complex plane
2nd week: function of complex variable, equations and inequalities with complex numbers
3rd week: Riemann surfaces
4th week: Sequences and series of complex numbers, Cauchy-Riemann conditions
5th week: Taylor and Laurent series (basic problems, classification of singularities, Picard's theorem)
6th week: Cauchy's integral formula, residue theorem, integration of complex functions (basic problems)
7th week: integration of complex functions (rational functions, semicircle contour)
8th week: integration of complex functions (multivalued functions, rectangular contour, special cases, gamma functions)
9th week: differential equations (basic problems, differential equations of first order, separation of variables, homogeneous differential equations)
10th week: differential equations of first order (exact equations, Euler multiplier)
11th week: linear differential equations of first order with examples from physics
12th week: linear differential equations of second order (method of variation of parameters)
13 th week: systems of differential equations
14 th week: Frobenius method (basic problems)
15 th week: Frobenius method (advanced problems)
GRADING AND ASSESSING THE WORK OF STUDENTS:
There are two "shorter" written exams during the semester which are not obligatory and which enable students to a) pass the written part of the exam and b) get some bonus points (5 points per completely solved problem; 20 bonus points max.) for the "standard" written exam. In a case when both of these "shorter" written exams are passed, a student can attend the oral part of the exam. Otherwise, a student has to pass the "standard" written exam in order to attend the oral part. The final grade is the average of the two grades (in written and oral part).
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- Butkov: Mathematical Physics (Addison-Wesley, 1968.)
I. Smolić: skripta za kolegije Matematičke metode fizike 1 i 2 (dostupna u pdf formatu na stranici predavača)
S. Benić, I. Smolić: skripta rješenih zadataka iz Matematičkih metoda fizike 1 i 2 (dostupna u pdf formatu na stranici asistenta)
G.B. Arfken, H. J. Weber: Mathematical Methods for Physicists (Academic Press, 1995.), Lang: Complex Analysis (Springer, 2003.), Tenenbaum, Pollard: Ordinary Differential Equations (Dover, 1985.)
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