Complex Analysis
Teacher
MR. Sasi Bhusan Padhi
Category
Core Courses
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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
- Text Book:
- 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:
- Complex Analysis – L. V. Ahlfors, Mc-Graw Hill International Editions (Vikas Publications – Second Edition)
Session Plan
Session 1
Session 2
Session 3
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 12
Functions of complex variable
https://www.youtube.com/watch?v=hrV-2TqF_ec
https://www.slideshare.net/niravbvyas/complex-function
Session 13
Differentiation of complex functions
https://www.youtube.com/watch?v=zuwObgLTda8
https://www.slideshare.net/hishamalmahsery/complex-numbers-and-functions-complex-differentiation
Session 14
Cauchy-Riemann Equations
https://www.youtube.com/watch?v=BLUV5uYaCxI
https://www.slideshare.net/sajidpk92/cauchy-riemann-equations
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 20
Session 21
Session 22
Session 23 & 24
Project-3
Finding conjugate harmonics of different harmonic functions and the corresponding analytic functions
Session 25
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
Definite integrals of complex functions
https://www.slideshare.net/AmitAmola/integration-in-the-complex-plane-52008858
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
Cauchy's Integral Formula
https://www.youtube.com/watch?v=gT9pK5OkM9Q
https://www.slideshare.net/HanpenRobot/cauchys-integral-formula
Session 46
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 58
Session 59
Session 63 & 64
Project-11
Study on Bilinear Transformations
Session 65 & 66
Project-12
Study on Conformal mappings
Case Studies
Case Studies
Our Main Teachers
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 […]