The aims of this course are to gather more knowledge regarding fundamental concepts, techniques and theories of complex algebra, calculus and geometry that are frequently used in real life data or in any branch of science.

After completing this course, the students should

learn domain, range, limit, continuity and differentiability of a function along with the categories of functions;

learn to formulate, identify and solve differential equations using several methods under boundary conditions or initial conditions.

learn techniques of complex analysis that make practical problems easy;

improve the ability to work with mathematical problems;

On successfully completion of this course, the student will be able to

know the functions of complex variable and concepts of continuity, differentiability and analyticity of such functions;

the ability to analyze data with appropriate mathematical tools and techniques;

develop the ability to think critically by verifying mathematical conjectures and launching theorems from complex analysis.

Complex number: Introduction, Properties of complex numbers, Differences with real number.

Complex functions: Different functions, limit and continuity, Complex differentiation and Cauchy Riemann equations.

Complex integration: Cauchy’s integral, Morera’s, Liouville’s, Rouches’s. Taylor’s, Laurant’s and Residue theorems. Evaluation of integrals, Elementary conformal transformations, Characteristic functions.

Transformations: Fourier, Hilbert and Wavelets transforms, and their applications.

1)

Fokas, A. S. (2003). Complex variables: introduction and applications. Cambridge University Press.

2)

Lipschutz, S., J. J. Schiller, & D. Spellman (2009). Schaum’s Outlines: Complex Variables: with an Introduction to Conformal Mapping and Its Applications. McGraw-Hill.

3)

Fisher, S. D. (1999). Complex variables. Courier Dover Publications.

4)

Polyanin, A. D., & A. V. Manzhirov (2006). Handbook of mathematics for engineers and scientists. CRC Press.

5)

Ponnusamy, S., & H. Silverman (2006). Complex variables with applications. Boston: Birkhäuser. [Wunsch]

6)

Poularikas, A. D. (2009). Transforms and applications handbook. CRC press.