Applied Mathematics
Course Objective
This course focuses on several branches of applied mathematics. The students are exposed to complex variable theory and a study of the Fourier and Z‑Transforms, topics of current importance in signal processing. The course concludes with studies of the wave and heat equations in Cartesian and polar coordinates.
 Complex Analysis(18 hours)
 Complex Analytic Functions
 Functions and sets in the complex plane
 Limits and Derivatives of complex functions
 Analytic functions. The Cauchy –Riemann equations
 Harmonic functions and it’s conjugate
 Conformal Mapping
 Mapping
 Some familiar functions as mappings
 Conformal mappings and special linear functional transformations
 Constructing conformal mappings between given domains
 Integral in the Complex Plane
 Line integrals in the complex plane
 Basic Problems of the complex line integrals
 Cauchy’s integral theorem
 Cauchy’s integral formula
 Supplementary problems
 Complex Power Series, Complex Taylor series and Lauren series
 Complex Power series
 Functions represented by Power series
 Taylor series, Taylor series of elementary functions
 Practical methods for obtaining power series, Laurent series
 Analyticity at infinity, zeros, singularities, residues, Cauchy's residue theorem
 Evaluation of real integrals
 The ZTransform(9 hours)
 Introduction
 Properties of ZTransform
 Z transform of elementary functions
 Linearity properties
 First shifting theorem, Second shifting theorem, Initial value theorem
 Final value theorem, Convolution theorem
 Some standard ZTransform
 Inverse ZTransform
 Method for finding Inverse ZTransform
 Application of ZTransform to difference equations
 Partial Differential Equations(12 hours)
 Linear partial differential equation of second order, their classification and solution
 Solution of one dimensional wave equation, one dimensional heat equation, two dimensional heat equation and Laplace equation(Cartesian and polar form) by variable separation method
 Fourier Transform(6 hours)
 Fourier integral theorem, Fourier sine and cosine integral; complex form of Fourier integral
 Fourier transform, Fourier sine transform, Fourier cosine transform and their properties
 Convolution, Parseval’s identity for Fourier transforms
 Relation between Fourier transform and Laplace transform
References:
 E. Kreyszig, “Advance Engineering Mathematics”, Fifth Edition, Wiley, New York.
 A. V. Oppenheim, “DiscreteTime Signal Processing”, Prentice Hall, 1990.
 K. Ogata, “DiscreteTime Control System”, Prentice Hall, Englewood Cliffs, New Jersey, 1987.
Evaluation Scheme
The questions will cover all the chapters of the syllabus. The evaluation scheme will be as indicated in the table below:
Chapter 
Hour 
Marks Distribution* 
1 
18 
30 
2 
9 
20 
3 
12 
20 
4 
6 
10 
Total 
45 
80 
*Note: There may be minor deviation in marks distribution.
