Method of Mathematical Physics PHYS3103

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%



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