Complex Analysis

Teacher

MR. Sasi Bhusan Padhi

Category

Core Courses

Course Attendees

Still no participant

Course Reviews

Still no reviews

Course Name : Complex Analysis

Code(Credit) : CUTM1520(3-1-2)

Course Objectives

  • To understand the application of Complex Analysis to Two-Dimensional problems in Physics including Hydrodynamics and Thermodynamics and also in Engineering fields such as; Nuclear, Aerospace, Mechanical and Civil engineering, signal processing & communications.
  • To acquire the skill of evaluating contour integrals using Cauchy's integral formula and Cauchy's integral theorem.

Learning Outcomes

Upon successful completion of this course, students will be able to:

 

  • To get equipped with the understanding of the fundamental concepts of functions of a complex variable along with the concepts of analyticity, Cauchy-Riemann relations and harmonic functions.
  • Evaluate complex contour integrals applying the Cauchy's integral theorem and Cauchy's integral formula.

Course Syllabus

Module I    ( T-5 hrs-P-4 hrs-P-2 hrs )

 

Complex numbers, Algebra of complex numbers, Modulus and argument, Roots of a complex number.

 

Practice-1  :  Printing a complex number and it's real and imaginary parts using Python

Practice-2  : Addition, Substraction, Multiplication and Division of two complex numbers using Python

Practice-3  :  Finding out the square roots of a complex number using Python

Practice-4  :  Finding out the argument of a complex number using Python

 

Project-1  :  Finding roots of different complex numbers

 

 

Module II    ( T-3 hrs-P-2 hrs-P-2 hrs )

 

 Functions of a complex variable, Limit and continuity, Differentiation, Cauchy-Riemann equations.

 

Project-2  :   Verifying Cauchy-Riemann equations for complex functions in cartesian form and polar form

 

 

Module III    ( T-4 hrs-P-0 hrs-P-2 hrs )

 

Analytic functions, Laplace equation, Harmonic and Conjugate harmonic functions, Method of  finding out conjugate harmonic functions and the corresponding analytic function, Entire functions.

 

Project-3  :   Finding conjugate harmonics of different harmonic functions and the corresponding analytic functions

 

 

Module IV    ( T-4 hrs-P-2 hrs-P-4 hrs )

 

Complex exponential functions, Logarithmic functions, Trigonomertric functions, Hyperbolic functions.

 

Project-4  :  Study on Complex Exponential and Logarithmic functions

Project-5  :   Studying properties of Complex trigonometric and hyperbolic functions

 

 

Module V    ( T-3 hrs-P-2 hrs-P-4 hrs )

 

Definite integrals of complex functions, Contours, Evaluation of Contour integrals using Cauchy's Integral Theorem.

 

Practice-5  :

 

Project-6  :   Evaluation of different types of Contour integrals using Cauchy's Integral Theorem

 

 

Module VI    ( T-6 hrs-P-2 hrs-P-6 hrs )

 

Cauchy’s Integral Formula for complex functions and their derivatives, Convergence of sequences and series, Taylor series, Maclaurin series.

 

Practice-6  :

 

Project-7  :  Evaluation of different types of Contour integrals using Cauchy's Integral Formula

Project-8  :  Study on Radius of convergence of a Power series

Project-9  :  Taylor series expansion of different complex functions

Project-10  :   Maclaurin series expansion of different complex functions

 

 

Module VII    ( T-5 hrs-P-0 hrs-P-4 hrs )

 

Conformal mapping, Bilinear Transformation, Fixed points, Cross ratio.

 

Project-11  :   Study on Bilinear Transformations

Project-12  :   Study on Conformal mappings

 

 

 

  1. Text Book:
    1. James Ward Brown and Ruel V. Churchill, Complex Variables And Applications, 8th Ed., Mc-Graw Hill International Edition, 2014.

    Chapters:  1, 2(12, 13, 15 to 26), 3(29, 30, 34, 35), 4(37 to 41,  50 to 52), 5(55, 56, 57), 8(93, 94), 9(101).

     

    Reference Book:

    1. Complex Analysis – L. V. Ahlfors, Mc-Graw Hill International Editions (Vikas Publications – Second Edition)

     

Session 6 & 7

Practice-1

Printing a complex number and it's real and imaginary parts using Python

Session 8 & 9

Practice-2

Addition, Substraction, Multiplication and Division of two complex numbers using Python

Session 10 & 11

Project-1

Finding roots of different complex numbers

Session 13

Differentiation of complex functions

https://www.youtube.com/watch?v=zuwObgLTda8

 

Session 14

Session 15 & 16

Practice-3

Finding out the square roots of a complex number using Python

Session 17 & 18

Project-2

Verifying Cauchy-Riemann equations for complex functions in cartesian form and polar form

Session 19

Session 22

Session 23 & 24

Project-3

Finding conjugate harmonics of different harmonic functions and the corresponding analytic functions

Session 26

Complex Logarithmic Function

https://www.youtube.com/watch?v=YUmMXPE3Mpo

Session 28

Complex Hyperbolic Functions

https://www.youtube.com/watch?v=eSNXRn91pKk

Session 29 & 30

Practice-4

Finding out the argument of a complex number using Python

Session 31 & 32

Project-4

Solving Complex Exponential and Logarithmic equations

Session 33 & 34

Project-5

Studying properties of Complex trigonometric and hyperbolic functions

Session 35

Session 37

Session 38 & 39

Practice-5

Session 40 & 41

Project-6

Evaluation of different types of  Complex integrals

Session 42 & 43

Project-7

Evaluation of different types of Contour integrals

Session 44

Session 45

Session 47

Radius of convergence of a power series

https://www.youtube.com/watch?v=YxahuwnVniI

Session 50 & 51

Practice-6

Session 52 & 53

Project-8

Study on Radius of convergence of a Power series

Session 54 & 55

Project-9

Taylor series expansion of different complex functions

Session 56 & 57

Project-10

Maclaurin series expansion of different complex functions

Session 60

Special Linear Fractional Transformations

https://www.youtube.com/watch?v=VJTxv-tRKj0

Session 63 & 64

Project-11

Study on Bilinear Transformations

Session 65 & 66

Project-12

Study on Conformal mappings

Case Studies

Case Studies

Recent Comments

    Our Main Teachers

    MR. Sasi Bhusan Padhi

    Asst. Prof. Department of Mathematics
    VIEW PROFILE

    MR. Sasi Bhusan Padhi works as an Assistant Professor, Department of Mathematics, Centurion University of Technology and Management. He has received M.Sc degree in Mathematics from Berhampur University in the year 2001 with Gold Medal. He has 17 years of teaching experience in different Graduate and Post Graduate colleges. Now he is doing research in […]