Complex Analysis & Numerical Methods

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MR. Sasi Bhusan Padhi

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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

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 integral theorem, Cauchy integral formula.
  • 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.

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.

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    MR. Sasi Bhusan Padhi

    Asst. Prof. Department of Mathematics
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    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 […]