Applied Mathematics

Course objectives
This course focuses on several branches of applied mathematics. The student is 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 diffusion equations in cartesian, cylindrical and polar coordinates.

1. Complex Variables (10 hours)
1.1 Function of Complex Variables.
1.2 Taylor series.
1.3 Laurent series.
1.4 Singularities, Zeros and poles.
1.5 Complex integration
1.6 Residues.

2 Z-Transforms (8 hours)
2.6 Linear, time invariant systems, response to the unit spike
2.7 Delay, advance, convolution
2.8 Definition of the Z-transform
2.9 Relation of convolution to the product of transform
2.10 Region of convergence, relationship to causality
2.11 Inverse of the Z-transform by long division and by partial fraction expansion
2.12 Parseval’s theorem

3 The Fourier integral (8 hours)
3.1 The Fourier integral
3.2 The inverse Fourier integral formula.
3.3 Frequency and phase spectra.
3.4 The delta function.

4 Partial differential equations (10 hours)
4.1 Basic concepts.
4.2 Wave equation.
4.3 Diffusion equation.
4.4 The Laplace equation in 2 and 3 dimensions.
4.5 Polar coordinates.
4.6 Cylindrical coordinates.
4.7 Spherical coordinates.
4.8 Bessels and Legendre equations.

5. Linear Programming (9 hours)
5.1 The simplex method.
5.2 The canonical forms of solutions.
5.3 Optimal values.

Textbook:
1.0 E. Kreyszig, “Advanced Engineering Mathematics”, Fifth Edition, Wiley, New York.

Reference for Z-Transform:
1.0 A.V. Oppenheim, “Discrete-Time Signal Processing”, Prentice Hall, 1990.
2.0 K. Ogota, “Discrete-Time Control Systems”, Prentice Hall, Englewood Cliffs, New Jersey,