Physics 725
Philosophy of the course
This is a course that contains way too much material for one semester, so we shall have to proceed expeditiously. I will cover the subjects in the order listed in the Syllabus below, at a rate determined by class comprehension. That is, I do not intend either to bore anyone or to leave anyone behind.
Here are the prerequisites for this course: calculus (through multivariate calculus—see item #2 in the syllabus below), ordinary differential equations, and some linear algebra (systems of simultaneous linear equations). To get a better idea of the class level and of what subjects will need emphasis, I will post an
on-line survey exam for you to do before the second class.
Questions are always welcome—I want students to provide rapid feedback.
Waiting for the results of HW and exams is too slow.
I plan to work examples in class. However, this is a hard subject that can only be mastered through long hours solving HW problems. In other words, watching me do problems on the blackboard will not get you through the course. I will thus try to schedule a 1 hour per week problem recitation session at a mutually agreeable time.
Basis of Course Grade
There will be bi-weekly homework assignments, a midterm and a final exam. The grade weightings will be as follows:
| |
Weighting for final grade |
| |
Item |
|
Weight |
| |
HW
(biweekly, about 6 assignments) |
|
30% |
| |
Midterm exam
(in class) |
|
25% |
| |
Final Exam
(in class) |
|
45% |
Syllabus
1. Infinite sequences, series and products
- Infinite sequences
- Infinite series
- Infinite Products
- Transformation of series
2. Review of calculus
- Functions
- Integration
- Differentiation
- Calculus-based inequalities
- Multivariate calculus
- Partial derivatives
- Vector analysis: grad, div and curl
- Useful identies
- Multivariate integration
- Curvilinear coordinates in 3 dimensions
3. Theory of Functions of a Complex Variable
- The complex number field
- Complex functions
- Continuity and analyticity
- Power series
- Elementary transcendental functions
- Contour integration
- Cauchy's Theorem
- Cauchy's integral formula
- Taylor's theorem
- The calculus of residues
- Singularities of analytic functions
- The number of zeros of an analytic function
- Rouche's Theorem
- Inverse functions and reversion of series
- Dispersion relations
4. Ordinary differential equations
- First order equations
- Linear differential equations
- Variation of parameters
- Power-series solutions
- Treatment of irregular singularities
5. Special functions
- Gamma and beta functions
- Legendre functions
- Associated Legendre functions
- Bessel Functions
- Hypergeometric functions
- Confluent hypergeometric function
- Special cases of hypergeometric functions
- Mathieu functions
6. Asymptotic approximations
- Asymptotic series
- WKBJ method for second order equations
- Stationary phase and steepest descents approximations
7. Linear vector spaces
- Linear equations, matrices and determinants
- Abstract Hilbert space
- The space of square-integrable functions L^2
- Complete orthonormal systems in L^2
- Fourier and Hankel transforms
8. Linear operators on Hilbert space
- Self-adjoint and normal linear operators
- Sturm-Liouville equations
- Compact linear operators
- Perturbation theory
9. Integral equations
- Solvability
- Tricks
- Fredholm method
- Reduction to finite system of linear equations
10. Partial differential equations
- Green's functions
- Integral transform methods