Complex Analysis & Numerical Methods

Course 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 5Evaluation 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 6Finding out Numerical solutions of differential equations using Runge-Kutta 2nd & 4th order methods

Text Books:

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:

Advanced Engineering Mathematics by P.V. O’Neil Publisher: Thomson
Fundamentals of Complex Analysis (with Applications to Engineering and Science) by E.B. Saff & A.D. Snider Publisher: Pearson
Numerical Methods for Scientific and Engineering Computation by M. K. Jain, S. R. K. Iyengar & R.K. Jain; New Age International Publishers.
Introductory Methods of Numerical Analysis by S.S. Sastry; Third Edition, Prentice Hall India.

Session Plan:

Session	Topic	Reference Link (if any)
1	Analytic Functions and Cauchy-Riemann Equations	Video Slides
2	Harmonic and Conjugate Harmonic Functions	Video MIT Notes Slideshare
3	Cauchy's Integral Theorem	Video 1 Video 2
4 & 5	Project-1 Verification of Cauchy-Riemann equations in Cartesian & Polar forms	—
6	Cauchy's Integral Formula	Video 1 Video 2 Video 3
7	Cauchy's Integral Formula for Higher Order Derivatives	Video NTNU PDF
8	Taylor Series	Video Slides
9 & 10	Project-2 Evaluation of contour integrals using Cauchy's Integral Formula	—
11	Laurent Series	Video 1 Video 2
12	Singularities	Video Slides
13	Residues	Video 1 Video 2
14	Residue Theorem & Evaluation of Type-I Integral	Video 1 Video 2
15	Interpolation	Video Slides
16	Lagrange’s Interpolating Polynomial	Video Slides
17 & 18	Project-3 Finding value at an interior point using Lagrange interpolation	—
19	Forward and Backward Difference Operators	Video PDF
20	Newton's Forward Difference Interpolation	Video PDF
21	Newton's Backward Difference Interpolation	Video PDF
22 & 23	Project-4 Value estimation using Newton's formulas	—
24	Trapezoidal Rule	Video Slides
25	Simpson's One Third Rule	Video Slides
26 & 27	Project-5 Evaluate definite integrals using Trapezoidal & Simpson's rule	—
28	Runge-Kutta 2nd Order Method	Video PDF
29	Runge-Kutta 4th Order Method	Video PDF
30	Problems on Runge-Kutta 4th Order	PDF
31 & 32	Project-6 Numerical solutions of differential equations using RK methods	—