Mathematical Physics-III
Teacher
Dr.Pratibha Tripathi
Category
Core Courses
Course Attendees
Still no participant
Course Reviews
Still no reviews
Course Name : Mathematical Physics-III
Code(Credit) : CUTM 1490(3-2-1)
Course Objectives
- The main objective of this course is to familiarize students with a range of mathematical methods that are essential for solving advanced problems in theoretical physics.
- The laws of physics are often expressed through the relatively complex mathematical apparatus.
- This course is intended to give mathematical tools necessary for a better understanding of the later courses in physics such as classical electrodynamics, quantum mechanics, solid-state physics, and statistical physics.
Learning Outcomes
- Perform algebra of complex numbers
- Express analytic complex function as power series.
- Identify the isolated singularities of a function and determine the types of singularity
- Calculate the Laplace Transform of basic functions using the definition
- Able to solve complex integrals
- Use the residue theorem to compute some definite integrals
- Use Scilab, Matlab and Python to solve Physical as well as mathematical problems
Course Syllabus
MODULE-I:
Complex Analysis
Motivation and introduction to Complex analysis; Brief revision of complex no and their graphical representation; the triangular inequality. Polar coordinates; Euler's formula and complex exponentials, polar form; Roots of the complex number- nth root, De-Moivers theorem; Representing complex multiplication as matrix multiplication, examples; Numerical problems
Practice 1: Basics of Python
Practice 2: Basics of Python
Practice 3: Basics of Python
Assignment 1: Numerical problems on Complex number and variable
Assignment 2: To write a python program to find the ares of the circle
To write a python program to find the sum of an array
MODULE-II:
Analytic Function
The derivative preliminaries. Numerical problems; Limits and continuous functions. Properties of limits and continuous functions, limit involving infinity; Branch cut, branch point and branches. Numerical problems; Cauchy-Riemann equation. Numerical problems; Complex line integral. Cauchy Integral theorem;
Practice 4: Compute nth roots of unity for n=2,3,4....
Practice 5: Find the Fourier Transform of exp(-x2)
Assignment 3: Numerical problems
Singular points, poles; Cauchy Integral formula for functions; Cauchy Integral formula for derivatives. Numerical problems; Harmonic function; Complex line integral.Cauchy Integral theorem; Numerical problems; Singular points, poles; Cauchy Integral formula for functions; Cauchy Integral formula for derivatives. Numerical problems: Harmonic function
Practice 6: Integrate 1/(x2+1)numerically and check with computer
integration
Assignment 4: Numerical problems
MODULE-III:
Taylor and Laurent Series:
Finite and infinite geometric series, convergence of power series; Taylor series; Numerical problems: Singularities, types of singularities; Laurent’s series, examples;
Assignment 6: Numerical problems on Taylor and Laurent Series
MODULE-IV
Residues:
Poles and residues;Residue theorem; Examples of poles and residues; Examples of poles and residues; Application of Residue theorems; Numerical problems-Solution of some definite integral using Residue theorem
Assignment 7: Numerical problems on Poles, Residues..
Assignment 8: Numerical Problems on Residue theorem
MODULE –V
Fourier Transform
Introduction, Fourier Integral trans form, examples; Fourier transforms- sine and cosine transforms; Fourier transform properties; Inverse Fourier transform, examples; Fourier transform- infinite wave train, Dirac Delta function and Gaussian function; Convolution of Fourier transform, Parseval Identity. Numerical Problems; Application of Fourier transform to one dimensional wave equation
Practice 7: Compute the Fourier Transform of exp(-x2)using Python and compare with numerically calculated solution
Assignment 9: Numerical problems on Fourier Transform
MODULE-VI:
Laplace Transform;
Introduction. Laplace transform of elementary function; Properties of Laplace transform;; Examples LT of 1st and 2nd order derivative; Inverse Laplace Transform; Examples LT of 1st and 2nd order derivative; Numerical problems; LT of a unit step function, Dirac Delta function; Application of LT to 2nd order differential equation, Application to damped harmonic oscillator. Impulse function and Lcr circuits
Assignment 10: Find the solution of Damped Harmonic oscillator using Laplace Transform
Practice 8 : Solve Kirchoff's current law for any node of an arbitrary circuit using LT
Practice 9: Solve Kirchoff's voltage law for any node of an arbitrary circuit using LT
Practice 1o: Perform circuit analysis of a general LCR circuit using Laplace Transform
MODULE-VII:
Green’s Functions
Introduction. Greens function for the one-dimensional problem; Construction of Greens function; Non-homogeneous boundary value problem; Homogeneous Boundary condition. Sturm- Liouville problem; Eigenvalue and Eigen function expansion of Greens function; Parseval Formula- complete condition for the Eigen function
Practice 11: Derive the Green's function for the operator d2/dx2 with boundary condition y(0)=0 and y(1)=0
Practice 12: Find an appropriate Green's function for the equation y''+1/4y=f(x) with boundary condition y(0)=y(π)=0
PRACTICE: Using Python
Assignment 11: Numerical problems on Green's function
Assignment 12: Numerical problems on Green's function
Text Books:
1. Mathematical Methods for Physicists: Arfken, Weber, 2005, Harris, Elsevier
Reference Books:
- Mathematical Physics by B.S.Rajput, Pragati Edition.
- Mathematical Physics by Satyaprakash, S.Chand&Co.
- Introduction to Mathematical Physics by Charlie Harper.
- Mathematical Physics by H.K.Dass and Dr. Rama Verma, S.Chand Publication.
- Mathematical Physics by B.D.Gupta.
Online Source: http://ocw.mit.edu
https://www.slideshare.net/ShreyPatel16/complex-function-96985172
Session Plan
Motivation and introduction to complex analysis
https://www.slideshare.net/ShawonRahman3/complex-number-65172003
https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018/https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018/
Session 2:
| Triangular inequality and polar coordinates
Examples
https://www.youtube.com/watch?v=hqr1DtXXHpY https://www.slideshare.net/hishamalmahsery/complex-numbers-and-functions-complex-differentiation |
Session 3:
Euler's Formula and comple exponentials, complex root-nth root
https://www.slideshare.net/jalilalmumtaz/1-introduction-to-complex-numbers
Session 4
Practice 1
Basics of Python
Session 5
Practice 2
Basics of Python
Session 6
Practice 3
Basics of Python
Session-7
Roots of complex number, nth root=De Moiver's theorem
https://www.ck12.org/book/ck-12-trigonometry-second-edition/section/6.7/
Session 8
Practice 4
Compute the square root of unity for n=2,3,3.... using Python
Session 9
Practice 5
Compute the square root of (-5+12i) using Python.
Session 10
Representing Complex multiplication as matrix multiplication
Session 11
Assignment 1
Numerical problems
https://fac.ksu.edu.sa/sites/default/files/2016_complex_analysis_problems_solutions.pdf
Session 12
Assignment 2
Write a Python program to find the area of the circle
Write a Python program to find the sum of an array.
Session 13
Limits, derivative preliminaries, continuity of the function
http://math.arizona.edu/~lega/322/Spring07/Complex_Numbers_3_4_Handout.pdf
Session 15
Cauchy-Reimann Equations, Examples
Session 16
Assignment 3
Cauchy-Reimann Equations- Numerical Problems
http://www.math.mcgill.ca/gantumur/math381f12/practicemidtermsolutions.pdf
Session 17
Complex Integral , Cauchy Integral theorems, Examples
Session-18
Practice 6
Integrate 1/(x2+1)numerically and check with computer integration
Session-19
Assignment 4
Numerical problems on complex Integral
https://fac.ksu.edu.sa/sites/default/files/2016_complex_analysis_problems_solutions.pdf
Session 20
Singular points, Types of singularity, poles, Examples
Cauchy Integral formula
Session 21
Assignments 5
Numerical problems on
Singular points, Types of singularity, poles, Examples
Cauchy Integral formula
Session 22
Finite and Infinite Geometric series, Taylor's series, examples
Session 24
Assignment 6
Numerical problems on Taylor and Laurent's series
Session 25
Poles and Residues, Residue at a simple pole and multiple poles
Session 26
Session 27
Application of Residue theorems- solutions of some definite integrals
Session 28
Assignment 7
solutions of some definite integrals using residue theorem
Session 29
Assignment 8
solutions of some definite integrals using residue theorem
Session 31
Fourier transform, sine transform and cosine transform
Session 32
Properties of Fourier Transform
http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html
Session 33
Inverse Fourier Transform, examples
http://www2.math.uconn.edu/~troby/Math200S08/fourier-transform.pdf
Session 34
FT of Infinite wave train, Gaussian function, Dirac Delta function
https://kaba.hilvi.org/homepage/cg/texts/gaussianfourier.pdf
Session 35
Convolution theorem, Parseval Identity, One dimensional wave equation
Session 36
Practice 7
Compute the Fourier Transform of exp(-x2)
Session 37
Assignment 9
Apply the Fourier Transform to an infinite wave train, numerical problems
Laplace Transform of elementary functions and properties
https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx
Session 39
LT of 1st and 2nd Derivatives,Inverse Laplace Transform
https://tutorial.math.lamar.edu/classes/de/inversetransforms.aspx
Session 40
LT of unit step function, Dirac Delta function and Convolution theorem
Session 41
Application to 2nd order Differential equation- Damped harmonic oscillator, Simple electrical circuits
Session 42
Practice 8
Compute Kirchoff's current law for any node of an arbitrary circuit using LT
Session 43
Practice 9
Compute Kirchoff's voltage law for any node of an arbitrary circuit using LT
Session 44
Practice -10
Perform circuit analysis of a general LCR circuit using LT using Python/Matlab
Introduction, Construction of Green's Function
Session 46
Non homogeneous boundary value problems
http://www.ams.sunysb.edu/~jiao/teaching/ams501_fall11/notes/greensfunc.pdf
Session 47
Homogeneous Boundary value problems- Sturm Liouville's problem
http://www.math.iitb.ac.in/~siva/ma41707/ode7.pdf
Session 48
Eigen function expansion of Greens Function, Parseval Formula
Session 50
Numerical problems on Green's Function
Session 51
Practice 11
Derive the Green's function for the operator d2/dx2 with boundary condition y(0)=0 and y(1)=0 using Matlab/ Python
Session 52
Practice 12
Compute an appropriate Green's Function for the equation y"+1/4y=f(x) with boundary condition and compare with numerical solutions
Our Main Teachers