Complex Variables and Transforms

This course is an undergrad level course designed to be studied by any one having understanding of College Level Math and Calculus.

This course can be divided into two parts. First part is related to complex numbers, complex variables and functions. While the second part is related to Transforms and Series.

After successfully completing this course, student is expected to be able define, understand and describe complex number system, complex functions, complex variables, transforms and series. Student is also expected to be able to analyze problems involving functions of complex variables, transforms and series (Fourier, Laplace, z Transforms and Fourier Series), limits, continuity, differentiability.

Topics:

Introduction to Complex Number: Complex Variable, Argand’s Diagram, Modulus and Argument of a Complex Number, Polar Form, De Moivre’s Theorem.

Complex Functions: Analytical Functions, Harmonic and Conjugate, Harmonic Functions.

Cauchy-Riemann Equations : Line Integrals, Cauchy’s Theorem, Cauchy’s Integral Formula, Independence of Path, Two Methods of Integration.

Fourier Series and Transform: Fourier Series / Transform of periodic / non-periodic functions, Properties of Fourier Transform, Inverse Fourier Transform, Convolution Theorem.

Laplace Transform: Laplace Transform of elementary functions, Concept and properties of Region of Convergence (ROC), Properties of Laplace Transform, Inverse Laplace Transform, Convolution Theorem, Heaviside Expansion Formula, Solution of Ordinary Differential Equations by Laplace Transform.

z- Transforms: basics and few numerical problems.