# Complex Analysis & Numerical Methods

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# 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 4 & 5

Project-1

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

## Session 9 & 10

Project-2

Evaluation of contour integrals using Cauchy’s Integral Formula

## 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 31 & 32

Project-6

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