Advanced mathematical analysis

Objectives and outcomes

Using the knowledge acquired in Mathematical Analysis, students are introduced to the mathematical analysis of functions of several variables, basic types of integrals of functions of several variables, Fourier series, basics of complex analysis, as well as to Laplace and Z transformations and their applications. Upon completion of the course, students will adopt the basic concepts, as well as the most important theorems covered during the course. Moreover, they will be able to apply the gained knowledge to other courses.


Functions of several variables: continuity, partial derivatives, gradient, total differential, unconditional and conditional extremuma, Taylor’s formula, Jacobian. Multiple integrals and their applications. Curvilinear integrals by arc and coordinates, Green-Riemann theorem, independence of curvilinear integrals from the path of integration, application. Surface integrals over the surface and coordinates-Ostrogradsky’s theorem and Stokes’ theorem. Field theory-divergence, rotor, classification of vector fields. Fourier series. Fourier transform. Complex analysis: complex functions, differentiability, analytical functions and their properties, integration, Laurent order, Cauchy residual theorem and applications. Laplace and Z transformation-concept, properties, applications.

Practical classes

Computational exercises, solving tasks that follow the units covered in lectures.