物理应用的复杂分析

Complex Analysis with Physical Applications

Terrified of differential equations and special functions in graduate level physics?
Come along, this course is for you.

474 次查看
莫斯科国立科技大学
edX
  • 完成时间大约为 10
  • 高级
  • 英语
注:因开课平台的各种因素变化,以上开课日期仅供参考

课程概况

The course is for engineering and physics majors.
You will learn how to build the solutions of important in physics differential equations and their asymptotic expansions.

The main topics include:
1.    Introduction to asymptotic series.
2.    Special functions.
3.    Saddle point techniques.
4.    Laplace method of solving differential equations with linear coefficients.
5.    Stokes phenomenon.

The course instructors are active researchers in a theoretical solid state physics.  Armed with the tools mastered while attending the course, the students will have solid command of the methods of tackling differential equations and integrals encountered in theoretical and applied  physics and material science.

课程大纲

Week 1. Asymptotic series. Introduction 

Asymptotic series as approximation of definite integrals. 
Examples, optimal summation Taylor vs asymptotic expansions.

Week 2. Laplace-type integrals and stationary phase approximations

Zero term and full Laplace asymptotic series.
Asymptotics of Error and Fresnel integrals.

Week 3. Euler Gamma and Beta-functions, analytic continuation and asymptotics

Euler Gamma function: definition, functional equation and analytic continuation.
Hankel representation for Gamma-function. 
Beta and digamma functions.
Asymptotic expansions.
Application of Gamma functions for the computation of integrals.

Week 4. Saddle point approximation I 

Introduction to the method of saddle point approximation.
The search for optimal deformation of the contour.
Full asymptotic series.
Elementary applications of the saddle point approximation.

Week 5. Saddle point approximation II

Subtleties of a contour deformation.
Contribution of end points.
Higher order saddles.
Coalescent saddle and pole.

Week 6. Differential equations with linear coefficients. Laplace method I

Construction of the solution of the differential equations with linear coefficients in terms of Laplace type contour integrals.
Examples of solutions of second order differential equations
The general outline of the technique.

Week 7. Physical applications

1D Coulomb potential  
Harmonic oscillator, method 1
Restricted harmonic oscillator 
Harmonic oscillator, method 2

Week 8. Stokes Phenomenon in asymptotic series and WKB approximation in Quantum Mechanics

Solution of Airy's equation by asymptotic series.
WKB approximation for solution of wave equations.
Asymptotics of Airy's function in the complex plane.
Stokes phenomenon.

Week 9. Differential equations with linear coefficients. Laplace method II (higher order equations)

Solutions of the differential equations of higher order by Laplace method.
More complicated examples.
Killer problems

Week 10. Final Exam

预备知识

Good knowledge of real and basics of complex analysis, differential equations and general physics.

千万首歌曲。全无广告干扰。
此外,您还能在所有设备上欣赏您的整个音乐资料库。免费畅听 3 个月,之后每月只需 ¥10.00。
Apple 广告
声明:MOOC中国十分重视知识产权问题,我们发布之课程均源自下列机构,版权均归其所有,本站仅作报道收录并尊重其著作权益。感谢他们对MOOC事业做出的贡献!
  • Coursera
  • edX
  • OpenLearning
  • FutureLearn
  • iversity
  • Udacity
  • NovoEd
  • Canvas
  • Open2Study
  • Google
  • ewant
  • FUN
  • IOC-Athlete-MOOC
  • World-Science-U
  • Codecademy
  • CourseSites
  • opencourseworld
  • ShareCourse
  • gacco
  • MiriadaX
  • JANUX
  • openhpi
  • Stanford-Open-Edx
  • 网易云课堂
  • 中国大学MOOC
  • 学堂在线
  • 顶你学堂
  • 华文慕课
  • 好大学在线CnMooc
  • (部分课程由Coursera、Udemy、Linkshare共同提供)

© 2008-2020 MOOC.CN 慕课改变你,你改变世界