Complex Analysis, Transform Techniques and Optimization (1BMATEE301)

Complex Analysis, Transform Techniques and Optimization

Course Code 1BMATEE301 
Scheme 2025
Type of Course Applied Science Course (ASC) 
Semester III
Teaching Hours/Week (L:T:P) 42:28:0 
CIE Marks 50
Total Hours of Pedagogy L /T/P/(SL&TW): 42:28:0:50 
SEE Marks 50
Credits 4 
Total Marks 100
Examination type (SEE) Theory 
Exam Hours 3

Module-1

Complex Analysis: Review of a 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 by Milne-Thomson method, Problems on electric flux.

Complex integration: Line integral of a complex function, Cauchy’s theorem and Cauchy’s integral formula and problems.

Module-2

Fourier Series: Periodic functions, Euler’s Formula, Conditions for Fourier expansion (Dirichlet’s condition), Fourier series of functions i) having points of discontinuity, ii) with period 2𝜋, iii) with change of interval (arbitrary period) and iv) Even and odd functions. Half rang Fourier series. Practical harmonic analysis.

Module-3

Infinite Fourier transforms: Definition, Fourier transforms, Fourier sine, and cosine transforms. Inverse Fourier Transforms, inverse Fourier sine and cosine transforms. Problems.

Z-transforms: Definition, Standard Z-transforms, Damping, and shifting rules, Statements of initial value theorem and Final Value Theorem. Inverse Z-transform and applications to solve difference equations.

Module-4

Probability Distributions: Review of basic probability theory, Random variables-discrete and continuous. Probability distribution function, cumulative distribution function, Mathematical Expectation-mean and variance. Repeated Trials- Binomial, Poisson, Normal and Exponential distribution (without proofs for mean and SD).

Sampling Theory: Introduction to sampling distributions, standard error, Type-I and Type-II errors. Test of hypothesis for means, single proportions only. Student’s t-distribution, Chi-square distribution as a test of goodness of fit.

Module-5

Linear Programming Problems: Introduction to optimization techniques, formulation of the problems.

Introduction to Linear programming, graphical solution, solution by simplex and Big M method.

Textbooks

(i)Higher Engineering Mathematics, B.S. Grewal, 45th Edition, Khanna Publishers.

(ii)Advanced Engineering Mathematics, Erwin Kreyszig, 10th Edition, Wiley India.

(iii)Higher Engineering Mathematics, H.K. Dass and Er. Rajnish Verma, S. Chand Publications.

Reference books

(i)Complex Variables and Applications by James Ward Brown and Ruel V. Churchill, McGraw-Hill Education.

(ii) Fourier Series and Boundary Value Problems by Ruel V. Churchill and James Ward Brown.

(iii)Probability and Statistics for Engineers and Scientists by Ronald E. Walpole et al., Pearson Education.

(iv)Operations Research by Hamdy A. Taha, Pearson Education.

(v) Schaum's Outline of Complex Variables, McGraw-Hill.