Complex Analysis and Special Functions I 1100-2Ind04

The course is devoted to analytic functions of a single complex variable and selected special functions useful in mathematical physics. It is obligatory for students of the 2nd year of the individual study, but should be interesting and accessible for other students, in particular of 3rd year. The course will be continued in the summer semester (as an optional course).

Plan of the course:
1. Complex differentiation
2. Contour integrals and residues
3. Taylor and Laurent series
4. Conformal transformations and homographies
5. Multivalued functions.
6. Riemann sphere
7. Gamma function and its properties
8. Infinite products
9. Asymptotic series.
10. Gauss and Fresnell integrals and their applications.
11. Laplace and stationary phase method and their applications
12. Differential equations in complex domain and their singular points.
13. Confluent equation and function.
14. Bessel equation and function.
15. Hypergeometric equation and function.

Prerequisites:
Analysis ,
Algebra and geometry

Description by Jan Dereziński, May 2008