# COMPLEX ANALYSIS, PROBABILITY AND STATISTICAL METHODS(18MAT41)

## COMPLEX ANALYSIS, PROBABILITY AND STATISTICAL METHODS

#### Course Learning Objectives:

• To provide an insight into applications of complex variables, conformal mapping and special
functions arising in potential theory, quantum mechanics, heat conduction and field theory.
• To develop probability distribution of discrete, continuous random variables and joint
probability distribution occurring in digital signal processing, design engineering and
microwave engineering.

### Module-1

Calculus of complex functions: Review of function of a complex variable, limits, continuity, and
differentiability. Analytic functions: Cauchy-Riemann equations in Cartesian and polar forms and
consequences.
Construction of analytic functions: Milne-Thomson method-Problems.

### Module-2

Conformal transformations: Introduction. Discussion of transformations: Bilinear transformations- Problems.
Complex integration: Line integral of a complex function-Cauchy’s theorem and Cauchy’s integral formula
and problems.

### Module-3

Probability Distributions: Review of basic probability theory. Random variables (discrete and continuous),
probability mass/density functions. Binomial, Poisson, exponential and normal distributions- problems (No
derivation for mean and standard deviation)-Illustrative examples.

### Module-4

Statistical Methods: Correlation and regression-Karl Pearson’s coefficient of correlation and rank correlation
-problems. Regression analysis- lines of regression –problems.
Curve Fitting: Curve fitting by the method of least squares- fitting the curves of the form-

### Module-5

Joint probability distribution: Joint Probability distribution for two discrete random variables, expectation
and covariance.
Sampling Theory: Introduction to sampling distributions, standard error, Type-I and Type-II errors. Test of
hypothesis for means, student’s t-distribution, Chi-square distribution as a test of goodness of fit.

### MODULE PAPER SOLUTIONS Course Outcomes:

At the end of the course the student will be able to:
• Use the concepts of analytic function and complex potentials to solve the problems arising in
electromagnetic field theory.
• Utilize conformal transformation and complex integral arising in aerofoil theory, fluid flow
visualization and image processing.
• Apply discrete and continuous probability distributions in analyzing the probability models
arising in engineering field.
• Make use of the correlation and regression analysis to fit a suitable mathematical model for
the statistical data.
• Construct joint probability distributions and demonstrate the validity of testing the
hypothesis.

### Question paper pattern:

5. The question paper will have ten full questions carrying equal marks.
6. Each full question will be for 20 marks.
• There will be two full questions (with a maximum of four sub-questions) from each module.

#### Textbook

1 Advanced Engineering Mathematics E. Kreyszig John Wiley & Sons 10th Edition,2016
2 Higher Engineering Mathematics B. S. Grewal Khanna Publishers 44th Edition, 2017
3 Engineering Mathematics Srimanta Pal et al Oxford University Press 3rd Edition,2016

### Reference Books

1 Advanced Engineering Mathematics C. Ray Wylie, Louis C.Barrett McGraw-Hill 6 th Edition 1995
2 Introductory Methods of Numerical Analysis S.S.Sastry Prentice Hall of India 4th Edition 2010
3 Higher Engineering Mathematics B. V. Ramana McGraw-Hill 11th Edition,2010
4 A Text Book of Engineering Mathematics N. P. Bali and Manish Goyal Laxmi Publications 2014

### Web links and Video Lectures:

1. http://nptel.ac.in/courses.php?disciplineID=111
2. http://www.class-central.com/subject/math(MOOCs)
4. VTU EDUSAT PROGRAMME - 20