Complex Analysis & Numerical Methods
Teacher
MR. Sasi Bhusan Padhi
Category
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
Analytic Functions and Cauchy-Riemann Equations
https://www.youtube.com/watch?v=du7WKHhqWdE
Session 2
Harmonic function and Conjugate harmonic functions
https://www.youtube.com/watch?v=JQSC0lCPG24
Session 3
Session 4 & 5
Project-1
Verification of Cauchy-Riemann equations for complex functions in Cartesian form and Polar form
Session 7
Cauchy's Integral Formula for higher order derivatives
Session 9 & 10
Project-2
Evaluation of contour integrals using Cauchy’s Integral Formula
Session 11
Session 14
Residue Theorem and Evaluation of Type-I integral
Session 15
Session 16
Lagrange’s interpolating polynomial
https://www.youtube.com/watch?v=F5pJG4YfQxs
https://www.slideshare.net/ayushraj116/lagrange-interpolation
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 19
Forward difference and Backward difference operators
Session 20
Newton's Forward difference interpolation formula
Session 21
Newton's Backward difference interpolation formula
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 24
Session 25
Session 26 & 27
Project-5
Evaluation of real definite integrals using Trapizoidal rule and Simpson’s one third rule
Session 28
Runge-Kutta 2nd order method
https://www.youtube.com/watch?v=agDf4wRjamE
http://mathforcollege.com/nm/mws/gen/08ode/mws_gen_ode_txt_runge2nd.pdf
Session 29
Runge-Kutta 4th order method
https://www.youtube.com/watch?v=FCRW2CGpEx8
https://mathforcollege.com/nm/mws/gen/08ode/mws_gen_ode_txt_runge4th.pdf
Session 30
Problems on Runge-Kutta 4th order method
https://mathforcollege.com/nm/mws/gen/08ode/mws_gen_ode_txt_runge4th.pdf
Session 31 & 32
Project-6
Finding out Numerical solutions of differential equations using Runge-Kutta 2nd & 4th order methods.
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 […]
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 […]