# Mathematical Physics-III

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# 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:

1. Mathematical Physics by B.S.Rajput, Pragati Edition.
2. Mathematical Physics by Satyaprakash, S.Chand&Co.
3. Introduction to Mathematical Physics by Charlie Harper.
4. Mathematical Physics by H.K.Dass and Dr. Rama Verma, S.Chand Publication.
5. Mathematical Physics by B.D.Gupta.

Online Source: http://ocw.mit.edu

M3

https://math.libretexts.org/Bookshelves/Differential_Equations/Book%3A_Differential_Equations_for_Engineers_(Lebl)/6%3A_The_Laplace_Transform/6.1%3A_The_Laplace_Transform

https://www.slideshare.net/ShreyPatel16/complex-function-96985172

## complex analysis pdf

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

Practice 1

Basics of Python

Practice 2

Basics of Python

Practice 3

Basics of Python

## 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 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 16

Assignment 3

Cauchy-Reimann Equations- Numerical Problems

http://www.math.mcgill.ca/gantumur/math381f12/practicemidtermsolutions.pdf

## Session 17

Session-18

Practice 6

Integrate 1/(x2+1)numerically and check with computer integration

Session-19

## Session 20

Singular points, Types of singularity, poles, Examples

## Session 21

Assignments 5

Numerical problems on

Singular points, Types of singularity, poles, Examples

Cauchy Integral formula

https://math.mit.edu/~jorloff/18.04/notes/topic4.pdf

## Session 22

Finite and Infinite Geometric series, Taylor's series, examples

https://www.slideshare.net/manojsingh786/unit1-71367674

## Session 24

Assignment 6

Numerical problems on Taylor and Laurent's series

https://math.mit.edu/~jorloff/18.04/notes/topic7.pdf

## Session 25

Poles and Residues, Residue at a simple pole and multiple poles

https://www.slideshare.net/manojsingh786/unit1-71367674

## Session 27

Application of Residue theorems- solutions of some definite integrals

http://jitkomut.eng.chula.ac.th/ee202/residue.pdf

## Session 28

Assignment 7

solutions of some definite integrals using residue theorem

https://math.mit.edu/~jorloff/18.04/notes/topic9.pdf

## Session 29

Assignment 8

solutions of some definite integrals using residue theorem

FT pdf

## Session 31

Fourier  transform, sine transform and cosine transform

## 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

http://wwwf.imperial.ac.uk/~jdg/eeft3.pdf

## 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

Laplace Transform pdf

## 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

## Green's Function

Greens function

Introduction, Construction of Green's Function

## 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

https://www2.ph.ed.ac.uk/~mevans/amm/section11.pdf

## Session 49

Numerical Problems

MScGreensFunction

## 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

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