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- 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.

- 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

**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

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/

Triangular inequality and polar coordinates
https://www.youtube.com/watch?v=hqr1DtXXHpY https://www.slideshare.net/hishamalmahsery/complex-numbers-and-functions-complex-differentiation |

Euler's Formula and comple exponentials, complex root-nth root

https://www.slideshare.net/jalilalmumtaz/1-introduction-to-complex-numbers

Practice 1

Basics of Python

Practice 2

Basics of Python

Practice 3

Basics of Python

Roots of complex number, nth root=De Moiver's theorem

https://www.ck12.org/book/ck-12-trigonometry-second-edition/section/6.7/

Practice 4

Compute the square root of unity for n=2,3,3.... using Python

Practice 5

Compute the square root of (-5+12i) using Python.

Representing Complex multiplication as matrix multiplication

Assignment 1

Numerical problems

https://fac.ksu.edu.sa/sites/default/files/2016_complex_analysis_problems_solutions.pdf

Assignment 2

Write a Python program to find the area of the circle

Write a Python program to find the sum of an array.

Limits, derivative preliminaries, continuity of the function

http://math.arizona.edu/~lega/322/Spring07/Complex_Numbers_3_4_Handout.pdf

Cauchy-Reimann Equations, Examples

Assignment 3

Cauchy-Reimann Equations- Numerical Problems

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

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

Singular points, Types of singularity, poles, Examples

Cauchy Integral formula

Assignments 5

Numerical problems on

Singular points, Types of singularity, poles, Examples

Cauchy Integral formula

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

Assignment 6

Numerical problems on Taylor and Laurent's series

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

Application of Residue theorems- solutions of some definite integrals

Assignment 7

solutions of some definite integrals using residue theorem

Assignment 8

solutions of some definite integrals using residue theorem

Fourier transform, sine transform and cosine transform

Properties of Fourier Transform

http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html

Inverse Fourier Transform, examples

http://www2.math.uconn.edu/~troby/Math200S08/fourier-transform.pdf

FT of Infinite wave train, Gaussian function, Dirac Delta function

https://kaba.hilvi.org/homepage/cg/texts/gaussianfourier.pdf

Convolution theorem, Parseval Identity, One dimensional wave equation

Practice 7

Compute the Fourier Transform of exp(-x2)

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

LT of 1st and 2nd Derivatives,Inverse Laplace Transform

https://tutorial.math.lamar.edu/classes/de/inversetransforms.aspx

LT of unit step function, Dirac Delta function and Convolution theorem

Application to 2nd order Differential equation- Damped harmonic oscillator, Simple electrical circuits

Practice 8

Compute Kirchoff's current law for any node of an arbitrary circuit using LT

Practice 9

Compute Kirchoff's voltage law for any node of an arbitrary circuit using LT

Practice -10

Perform circuit analysis of a general LCR circuit using LT using Python/Matlab

Introduction, Construction of Green's Function

Non homogeneous boundary value problems

http://www.ams.sunysb.edu/~jiao/teaching/ams501_fall11/notes/greensfunc.pdf

Homogeneous Boundary value problems- Sturm Liouville's problem

http://www.math.iitb.ac.in/~siva/ma41707/ode7.pdf

Eigen function expansion of Greens Function, Parseval Formula

Numerical problems on Green's Function

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

Practice 12

Compute an appropriate Green's Function for the equation y"+1/4y=f(x) with boundary condition and compare with numerical solutions