Syllabus for Method of Mathematical Physics PHYS3103
Current Lecturer
Prof. Weixin LU Email: luweixin@suda.edu.cn
Course Time
3rd Semester
Lectures: 4 sessions/week, 50min /session. 18 weeks.
Course Description
Mathematical method for physicists is a fundamental course for many specialties. The course aims to provide students with: complex variables, analytic functions, Cauchy theorem, contour integrals, Taylor series, Laurent series, residues, ordinary differential equations, wave equations, diffusion equations, Laplace equations, the method of separation of variables, Legendre polynomial, Bessel functions, Fourier transforms and Laplace transforms. The course is a prerequisite to Electrodynamics, Statistical Physics, Solid State Physics and Quantum Mechanics.
Prerequisites
Calculus (00071012/13); Linear Algebra (00071004)
Textbooks
Philippe Dennery and Andre Krzywicki, Mathematics for physicists, Dover Publications, INC., Mineola, New York (chapter I & IV).
Main Contents
Week | Teaching Contents | Sessions | Objectives |
1 | complex number, functions of a complex argument, analytic function, Cauchy-Riemann conditions, harmonic function | 4 | understand the definition of complex number and complex function, the existence of the limitation and the condition of continuity, understand the condition of analytic functions, understand the definition of harmonic functions |
2 | simple analytic function, integral calculus of complex functions | 4 | understand the definition and properties of analytic functions, understand the method of the integral of complex functions |
3 | Cauchy theorem, the formula of Cauchy’s integral | 4 | understand and use the theorem of the contour integral of complex functions and the formula of Cauchy’s integral |
4 | exercise class, series of analytic functions | 4 | interpret the problems via exercises, understand the basic properties of series, familiar with the method to expand complex functions by series |
5 | Taylor series and Laurent series, types of singularity, residues | 4 | understand the method of expanding by Taylor series and Laurent series, familiar with the types of singularity, familiar with the definition and calculation of the residues |
6 | integral of real function by residues, exercise class | 4 | understand the method to calculate the integral of real functions using the theorem of residues, interpret the problems of exercise |
7 | midterm, ordinary differential equations | 4 | familiar with the solutions the ordinary differential equations |
8 | wave equations and its solutions | 4 | understand the method to obtain the mathematical equations and boundary conditions, the solutions of the homogeneous equations |
9 | solutions of inhomogeneous equations, diffusion equations and its solutions | 4 | familiar with the solutions of the inhomogeneous wave equations, understand the solutions of the diffusion equations |
10 | solutions of the 2D Laplace equation | 4 | understand the solutions of the 2D Laplace equation |
11 | D'alembert's solutions of the wave equations and the solutions of high dimensional wave equations | 4 | understand the solutions of the 1D wave equations with no boundary conditions, familiar with the solutions of high dimensional wave equations |
12 | exercise class, Legendre polynomial | 4 | understand the method of separation of variables to solve the 3D Laplace equations |
13 | primitive function of the Legendre polynomial, recurrence relations and expansion | 4 | understand the primitive function of the Legendre polynomial and its recurrence relations, familiar with the method to expand functions by Legendre polynomial |
14 | Associated Legendre polynomial, sphered Dirichlet problems | 4 | understand the solutions of 3D Laplace equations in spherical coordinate system |
15 | exercise class, Bessel function, Bessel equation | 4 | understand the solution of Bessel’s differential equations |
16 | primitive function of Bessel function, expand by Bessel functions | 4 | understand the primitive functions of the Bessel functions and their recurrence relations, and the method to expand functions by Bessel functions |
17 | Bessel function of the second and third kind, exercise class | 4 | understand the definition of the Bessel functions of the second and third kinds |
18 | Fourier transforms, Laplace transforms | 4 | understand the Fourier transforms, the definition of the Laplace transforms |
Marking Scheme:
Attendance | Homework | Midterm | Final exam |
10% | 10% | 20% | 60% |