Complex Analysis & Numerical Methods

Teacher

Dr. Sasi Bhusan Padhi

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Core Courses

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Course Name : Complex Analysis & Numerical Methods

Code(Credit) : CUTM1003(2-0-1)

Course Objectives

  • To understand about Complex variables and complex functions.
  • To acquire the skill of evaluating contour integrals using Cauchy's integral formula and Cauchy's integral theorem.
  • To understand the limitations of analytical methods and the need for numerical methods and the ability to apply these numerical methods to obtain the approximate solutions to engineering and mathematical problems.

Learning Outcomes

COs

Course Outcomes

Mapping with COs with POs

(High-H, Low-L, Medium-M)

CO1
Derive a variety of numerical methods
for finding out solutions of various
mathematical problems arising in roots
of linear and non-linear equations,
Solving differential equations with initial
conditions and Evaluating real definite
integrals.
 PO2(3)
CO2
To get equipped with the understanding of
the analytical and fundamental concepts
of functions of a complex variable along
with the concepts of analyticity, Cauchy-
Riemann relations and harmonic
functions
PO1 ( 3)
CO3
Solving complex contour integrals
applying the Cauchy integral theorem,
Cauchy integral formula.
PO2(3), PO1(2)

Course Syllabus

Module I    (T-3 hrs-P-0-hrs-P-0 hrs)

 

Functions of a complex variable, Analytic functions, Cauchy-Riemann equations (Without Proof), Harmonic and Conjugate harmonic functions, Cauchy’s Integral Theorem (Without Proof).

 

Project-1 : Verification of Cauchy-Riemann equations for complex functions in Cartesian form and Polar form

 

 

Module II    (T-3 hrs-P-0 hrs-P-2 hrs)

 

Cauchy’s Integral Formula (Without Proof), Cauchy’s Integral Formula for higher order derivatives (Without Proof), Taylor series.

 

Project-2 : Evaluation of contour integrals using Cauchy’s Integral Formula

 

 

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

 

Laurent series (Without Proof), Pole, Residue, Residue Theorem (Without Proof), Evaluation of Real integral Type-I.

 

 

 Module – IV    (T-2 hrs-P-0 hrs-P-2 hrs)

 

Interpolation, Lagrange interpolation polynomial.

 

Project-3 : Finding out the value of a given function at an interior point on an unequal interval using Lagrange interpolation polynomial

 

 

 Module – V    (T-3 hrs-P-0 hrs-P-2 hrs)

 

Forward and backward difference operators, Newton’s forward and backward difference Interpolation formulae.

 

Project-4 : Finding out the value of a given function at an interior point on an equal interval using Newton’s forward and backward difference interpolation formulae

 

 

Module – VI    (T-2 hrs-P-0 hrs-P 2 hrs)

 

Numerical Integration, Trapizoidal rule, Simpson’s one third rule.

 

Project-5 : Evaluation of real definite integrals using Trapizoidal rule and Simpson’s one third rule

 

 

Module – VII    (T-3 hrs-P-0 hrs-P-2 hrs)

 

Runge-Kutta 2nd & 4th order methods.

 

Project-6 : Finding out Numerical solutions of differential equations using Runge-Kutta 2nd & 4th order methods

 

 

 

Text Book:

 

1) Advanced Engineering Mathematics by E. Kreyszig Publisher: Johnwilley & Sons Inc-8th Edition Chapters : 12 (12.3, 12.4), 13 (13.2 to 13.4), 14.4, 15 (15.1 to 15.4 Only Type-I integral), 17 (17.3, 17.5), 19 (19.1).

 

 

Reference Books:

 

1) Advanced Engineering Mathematics by P.V. O’Neil Publisher: Thomson

2) Fundamentals of Complex Analysis (with Applications to Engineering and Science) by E.B. Saff & A.D. Snider Publisher: Pearson

3) Numerical Methods for Scientific and Engineering Computation by M. K. Jain, S. R. K. Iyengar & R.K. Jain; New Age International Publishers.

4) Introductory Methods of Numerical Analysis by S.S. Sastry; Third Edition, Prentice Hall India.

Session Plan

Session 1

Session 4 & 5

Project-1

Verification of Cauchy-Riemann equations for complex functions in Cartesian form and Polar form

Session 7

Session 9 & 10

Project-2

Evaluation of contour integrals using Cauchy’s Integral Formula

Session 14

Session 17 & 18

Project-3

Finding out the value of a given function at an interior point on an unequal interval using Lagrange interpolation polynomial.

Session 22 & 23

Project-4

Finding out the value of a given function at an interior point on an equal interval using Newton’s forward and backward difference interpolation formulae.

Session 26 & 27

Project-5

Evaluation of real definite integrals using Trapizoidal rule and Simpson’s one third rule

Session 30

Session 31 & 32

Project-6

Finding out Numerical solutions of differential equations using Runge-Kutta 2nd & 4th order methods.

Case Studies

Our Main Teachers

Dr. Sasi Bhusan Padhi

Asst. Prof. Department of Mathematics
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Dr. 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 2000 with Gold Medal and Ph. D from Centurion University of Technology and Management. He has more than 20 years of teaching experience in […]

 Professor ,School of Applied Sciences, Centurion University of Technology and Management, Paralakhemundi, Odisha, India. She has over 26 years of teaching and 8 years of research experiences. She has published 35 research articles in National and International refereed journals and has authored 01 book and 02 book chapters. She has produced 01 PhD scholar and […]