|Complex Analysis & Programming||Learning Schedule|
|Pre-requisites: Advanced Engineering Mathematics||3||1||0||4|
This course is an introduction to the concepts of Partial differential equations and their solution. The calculus of function of complex variable is discussed. Among the most important topics are Method of separation of variables and its applications to wave equation, one dimensional heat equation and two-dimensional heat flow, Analytic function, Cauchy-Riemann Equations, Harmonic functions with application to flow problem, Zeroes and Singularities of complex valued functions, Residues, Residue theorem and It’s application in evaluation of real integrals around unit and semi circle. Z-Transform is also introduced and applied in solving difference equation
To introduce the concepts of Partial Differential Equations that are often encountered in engineering study and techniques to solve them. To understand the behavior of complex variable function and calculus of complex variable functions. The introduction of Z-Transform and it’s application in solving difference equation gives an exposure of discrete transform to the students. Each of these tools has immense practical application and lays a foundation of various courses in their future course of study.
By the end of the course the students are expected to be able to:
- Understand concepts of Partial Differential Equations and techniques to solve them.
- Understand the behavior of complex variable function and importance of a special class of function, analytic functions in evaluating complex and real integrals.
- Understand the application of Z-Transform in solving difference equation.
Unit I: Linear Programming: Linear programming problems formulation, Solving linear programming problems using Graphical method, Simplex method and Dual simplex method.
Unit II: Functions of Complex Variable: Some standard functions of complex variables, Limit, Continuity, Differentiability of function, Analytic function, Cauchy-Riemann Equations in Cartesian and Polar form (with Proof), Sufficient conditions for a function to be analytic (without Proof), Harmonic functions.
Unit III: Complex Integration : Cauchy- Goursat theorm(Only statement and applications), Generalized Cauchy Integral formula (with-out Proof), Taylor’s and Laurent’s series (Without Proof), radius and circle of convergence, Zeroes and Singularities of complex valued functions, Residues, Residue theorem and it’s application in evaluation of real integrals around unit and semi circle.
Unit IV: Tests of Hypothesis and Significance
Hypothesis testing, Null and Alternate hypothesis, test of hypothesis and significance, Special tests of significance for Large samples and Small samples (F, chi- square, z, t- test).
Unit V: Probability and its Distributions: Conditional probability, Bayes theorem and its applications, expected value of a random variable. Properties and application of Binomial, Poisson and Normal distributions.
- Advanced Engineering Mathematics : K. Jain and S. R. K. Iyengar, Narosa Publishers.
- Advanced Engineering Mathematics : B.S. Grewal Khanna Publishers.
- Advanced Engineering Mathematics : Michael D. Greenberg, Pearson Education, Asian
- Advanced Engineering Mathematics : Kreyszig, John Wiley & Sons.