### Methods of Mathematical Physics

Course Code: B22003Y-02

Course Name: Methods of Mathematical Physics

Credits: 4.0

Pre-requisite: Advanced Mathematics, Mechanics, Electromagnetics, Optics

Lecture Time: 40 sessions, 2 hours/session

Course Description

Methods of Mathematical Physics is an important compulsory basic course for students majoring in physics, and is listed as a main basic course in the education program. The emphasis of this course is to develop students' ability to apply mathematical methods to solve practical physical problems, and lay the essential mathematical foundation for learning the follow-up professional courses and basic courses (including electrodynamics, quantum mechanics and solid physics, etc.). This course is designed not only to let students obtain knowledge, but also to put emphasis on improving the ability of students to build a physical model, as well as the ability to develop and innovate knowledge. Through the study of this course, students should meet the following basic requirements.

This course is divided into two parts, the complex variable function theory and mathematical physics equation.

Part 1: Through the study of the complex function theory, the students should fully understand some basic concepts and basic theories of the complex function, and master the skills, methods and rules of the complex operation; Understand the multivalue function, analytic function and Cauchy theorem and the residue theorem, etc; Master the method of calculating the definite integral of the real variable function; Understand the concept of Taylor series and Laurent series and the concept of technical convergence; Comprehend mathematical and physical significance of the Fourier transform.

Part 2: Learn the definite solution method of mathematical physics equation. The method of separation of variables is the common method of solving initial boundary value problem of various linear partial differential equations. Students should master the basic steps and methods of establishing mathematical models (partial differential equations) according to the specific physical process, and improve the ability to analyze and solve practical problems.

The specific content includes: understanding and mastering how to formulate physical problems as mathematical physics equations by infinitesimal method, and determine the boundary conditions; skillfully mastering the method of separation of variables of solving homogeneous equations under homogeneous and non-homogeneous boundary conditions; mastering the power series solution of the ordinary differential equation with non-constant coefficients; Understanding the basic properties of the axisymmetric  spherical function and the general spherical function, and grasp the method of the generalized Fourier series expansion based on the spherical function; getting to know the Bessel function, spherical Bessel function, Bessel function with imaginary argument, Neumann function and Hankel function and the  properties of the corresponding ordinary differential equations; solving ordinary differential equations and partial differential equations using integral transform method.

Teaching Features

1. Implement graphics visualization teaching to enhance the intuitive feeling;

2. Combined with the specific physical problems, improve the ability of students to analyze and establish mathematical models;

3. Through the study of mathematical knowledge, reveal the mathematical logic of the physical knowledge students mastered;

4. Inspire students to find correlation of physical knowledge of different subjects, and stimulate the intuitive imagination.

Topics and Schedule

Part 1. Complex Variable Function Theory(32 hrs)

Chapter 1. Complex Variable Function(7 hrs)

Main Content: Complex operation and complex variable function (1 hr); Derivative of complex variable function (1 hr); analytic function (2 hrs); multivalue function (1 hr); planar scalar field (1 hr); evolution of number (1 hrs)

Requirement: Beginning with the basic operations of complex number, introduce the complex function and some related mathematical concepts such as neighbourhood, limit, derivative; By graphic representation, we will enable students to understand the basic properties of analytic function. We put importance on the comparison of complex function with real function.

Key Point: Cauchy-Riemann condition, analytic function, relationship between real part and imaginary part of analytic function, geometric meaning of derivatives.

Difficult Point: visualization of complex variable function, multivalue function, Riemann surface.

Chapter 2. Integral of Complex Variable Function (4 hrs)

Main Content: Integral of complex variable function (1 hr); Cauchy theorem and indefinite integral (1 hr); Cauchy integral formula (2 hrs).

Requirement: path integral characteristic of complex variable function; application of Cauchy theorem and Cauchy integral formula; physical meaning of the Cauchy integral formula; relationship between analytic function and conservative potential.

Key Point: concept of regional and connectivity; understand physical meaning of the Cauchy integral formula.

Difficult Point: use the higher order derivative of Cauchy integral formula to calculate integral of complex variable function.

Chapter 3. Power Series Expansion (6 hrs)

Main Content: convergence of power series (1 hr); Taylor series expansion (1 hr); Taylor series expansion (1 hr); Laurent series expansion (2 hrs); analytic continuation (1 hr); classification of isolated singularities (1 hr).

Requirement: Learn and master the power series expansion of the complex variable function by comparing with power series expansion of the real variable function; Understand the calculation method of convergence radius of power series; Master the expansion of analytic functions into Taylor series and the expansion into Laurent series in different region.
Key Point: Discriminant method of convergence of power series; Taylor series expansion.

Difficult Point: Laurent series expansion; analytic continuation.

Chapter 4.  Residue Theorem (7 hrs)

Main Content: Residue theorem (1 hr); calculate real variable integral function by using the Residue theorem (2 hrs); Integration of multivalue functions (2 hrs); calculate the sum of series by using the Residue theorem (1 hr)*.

Requirement: The residue is one of the important concepts in complex variable function theory and this chapter is one of the important chapters of this course. Through examples, we will explain how to make use of the residue theorem to calculate the real integral function.

Key Point: Residue theorem and its application; Jordan lemma; calculation method of three kinds of real variable function.

Difficult Point: Selection of integral circuit; Integration of multivalue functions.

Chapter 5.  Fourier Transform (4 hrs)

Main Content: Fourier series (1 hr); Fourier integral and Fourier transform (2 hrs); δ function (1 hr).

Requirement: Master the periodic function, function defined on a finite interval, Fourier expansion of odd function and even function; Understand Fourier expansion, Fourier transform, and properties of non-periodic function; Master the common δ function in Physics, its properties and mathematical treatment method; Because Fourier series has been discussed in the Advanced Mathematical course, the relative part of this chapter can be omitted.
Key Point: Properties of Fourier series and Fourier transform; Physical meaning of Fourier transform

Difficult Point: δ function, convolution

Chapter 6.  Laplace Transform (4 hrs)

Main Content: Property of Laplace transform (2 hrs); Inversion and application of Laplace transform (2 hrs).

Requirement: Understand basic property of Laplace transform; Learn and master how to use Laplace transform to solve the initial value problem of ordinary differential equations.

Key Point: Property of Laplace transform; use Laplace transform to solve the initial value problem of ordinary differential equations.

Difficult Point: Inversion of Laplace transform.

Part 2. Mathematical Physical Equation(42 hrs)

Chapter 7. Derivation of mathematical physical equation and condition of the definite solution (8 hrs)

Main Content: continuous medium system and vibration equation (2 hrs); Heat conduction equation and diffusion equation (2 hrs); other two order linear partial differential equation and superposition principle (1 hr); definite solution condition (1 hr); D'Alembert formula and traveling wave method (2 hrs).
Requirement: The uppermost priority of this part is to learn how to transform the physical problem into a mathematical physics equation and students are required to have enough background physical knowledge of the relative physical problem. The important goal of this part is to express the background physical knowledge into mathematical form. Understand definite solution conditions, boundary conditions, cohesion conditions and D'Alembert formula of vibration equation and its physical meaning.

Key Point: Obtain the motion equation of continuous medium using infinitesimal method according to physical problem and physical law; Understand boundary condition and initial condition of special physical problems.

Difficult Point: Simplification of the physical problem to obtain the motion equation; Wave reflection and half wave loss.

Chapter 8.  Variable Separation Approach (8 hrs)

Main Content: Definite solution problem of homogeneous equation with homogeneous boundary condition (2 hrs); Definite solution problem of homogeneous equation with inhomogeneous boundary condition (2 hrs); Definite solution problem of inhomogeneous equation with homogeneous boundary condition (2 hrs); Poisson equation (1 hr); A summarize of variable separation approach (1 hr).

Requirement: Variable separation approach is one of the most basic methods to solve mathematical physics equation which has different treatment means to different situations with various General equations, boundary conditions, and (or) initial conditions.

Key Point: Two kinds of inhomogeneous: 1) inhomogeneous equation, 2) inhomogeneous boundary condition; Eigen function and eigenvalue; Fourier series expansion; Meaning of homogeneous boundary condition.

Difficult Point: Search of specific solutions; relationship of formal treatment methods of definite solution problems and specific skills for specific problems.

Chapter 9. Series solutions of two order linear ordinary differential equations (8 hrs)

Main Content: Ordinary differential equations of special functions (1 hr); A primary course of field theory (1 hr)*; Series expansion method for constant point neighborhood (2 hrs); Series expansion method for canonical singular point neighborhood (2 hrs); Sturm-Liouville eigenvalue problem (2 hrs).

Requirement: How the ordinary differential equations of various special functions are generated from the common mathematical physics equations and the relationships between all kinds of equations; understand spherical coordinate and cylindrical coordinate system; The series solution of ordinary differential equations in the neighborhood of the regular singular point and the canonical singular point, and the difference between these two kinds of series solution; Understand Sturm-Liouville eigenvalue problem.

Key Point: the characteristic of ordinary differential equations corresponding to special functions and its relation with coordinate system; series expansion method.
Difficult Point: the series solution of ordinary differential equations in the neighborhood of the canonical singular point.

Chapter 10.  Spherical Function (6 hrs)

Main Content: Legendre polynomials and axisymmetry problem in spherical coordinate systems (2 hrs); Associated Legendre function (2 hrs); Spherical harmonic function and its application to non-axisymmetric problems (2 hrs).

Requirement: Legendre polynomials; recursive relation; generating function; associated Legendre function; spherical harmonic function; generalized Fourier series and its application.

Key Point: the properties and applications of Legendre polynomials; generalized Fourier series expansion with special functions.
Difficult Point: Orthogonality of generalized Fourier series; The expansion coefficients of associated Legendre function and spherical harmonics function.
Chapter 11.  Cylinder Function (8 hrs)

Main Content: Bessel function, Newmann function and its properties (2 hrs); Eigenvalue problem of the integer order Bessel equation and its application (2 hrs); Hankel function and virtual Bessel function (2 hrs); Sphere Bessel function (1 hr); Series and the method of generating function (1 hr)*.

Requirement: Bessel function, Newmann function and its properties including orthogonality and asymptotic behavior; Generating function;Application of cylinder function in physics problems.

Key Point: Basic properties and application of Bessel, etc.
Difficult Point: Determine methods of obtaining the eigenvalues of different boundary conditions; Generalized Fourier series expansion coefficient, etc.

Chapter 12.  Integral Transform Method (4 hrs)

Main Content: Using the Fourier transform method to solve the problem of infinite space (2 hrs); Using Laplace transform method to solve the initial value problem, and the combination of the two methods (2 hrs).
Requirement: Know application scope of Fourier transform and Laplace transform method and master the method for inverse transformation.
Key Point: basic properties and applications of the integral transformation method.

Difficult Point: Application of inverse transform and convolution.

Midterm Examination (2 hrs)
Examination form: closed-book written examination
Composition of final score:
Midterm Examination (the part of complex variable function) 30%

Final Examination (the part of partial differential equations and special functions) 40%

Daily Performance (attendance and homework)30%

Textbook

Kun-Miao Liang, Methods of Mathematical Physics, (the Fourth Edition), Higher Education Press, 2010

References

 H. Jeffreys and B. Jeffreys, Methods of Mathematical Physics, Third Edition, World Book Inc, 2002
 Tai L. Chow, Mathematical Methods for Physicist: A Concise Introduction, Cambridge University Press, 2000

 Chong-Shi Wu, Methods of Mathematical Physics, Second Edition, Peking University press

 Duan-Zheng Yao, Methods of Mathematical Physics, Third Edition, Science Publishing Company