Mathematics III L 1102-201L

Program:

I) Elements of differential geometry in R³
1) Curves
- Sarret-Frenet equations
2) Surfaces
3) Curvilinear coordinates
4) Gradient, divergence, rotation and laplacian
5) Separation of variables for partial differential equations
II) Line and surface integrals
1) Gauss' theorem
2) Stokes' theorem
3) Elements of differential forms
III) Complex analysis of one variable
1) Functions of a complex variable
- The Cauchy-Riemann relations
- Power series in a complex variable
- Multivalued functions and branch cuts
- Singularities and zeros
- Conformal transformations
2) Complex integrals
- Cauchy's theorem
- Cauchy's integral formula
3) Taylor and Laurent series
4) Analytic continuation
5) Residue theorem with applications
6) Euler's Gamma and Beta functions
IV) Fourier series and transforms
1) Fourier series
- Dirichlet's theorem
- Parseval's theorem
2) Fourier transform
- Parseval's theorem
- Applications to linear partial differential equations
3) Elements of distribution theory
4) Laplace transform
- Applications to linear ordinary equations
V) Orthogonal polynomials
1) Legendre, Laguerre and Hermite polynomials
VI) Analytic theory of the second order ordinary differential equations
1) Series solutions
2) Hypergeometric and Confluent hypergeometric equations
3) Bessel and spherical Bessel functions
VII) Eigenfunction methods for differential equations
1) Hermitian differential operators
2) Sturm-Liouville equations
3) Superposition of eigenfunctions and Green's functions

http://www.fuw.edu.pl/~jkam
http://www.fuw.edu.pl/~jkam/Dydaktyka/Matematyka/Matematyka3L2008_2009.htm

Description by Jerzy Kamiński, June 2008