# Complex Analysis

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# 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 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 23 & 24

Project-3

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

## Session 26

Complex Logarithmic Function

## Session 28

Complex Hyperbolic Functions

## 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

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

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

## Session 63 & 64

Project-11

Study on Bilinear Transformations

## Session 65 & 66

Project-12

Study on Conformal mappings