KOMPLEKSNA ANALIZA 1

After passing the exam from the subject Complex Analysis I it is expected that the students are able: 1. To define the complex number in the algebraic and in the trigonometric form, to define the operations over complex numbers, to prove their properties and to present a geometric interpretation. 2. To define the metric on the sets C and Ĉ. 3. To define a sequence of complex numbers, the convergence of a sequence and to prove the basic properties of convergent sequences. 4. To define the elementary functions (power, polynomial, rational, exponential, trigonometric, hyperbolic, logarithmic, root, inverse trigonometric and inverse hyperbolic functions), and to prove their properties. 5. To define the differentiability of functions of complex variables. To define the harmonic function. To prove the basic properties. 6. To define the integral of a complex function. To formulate and to prove Cauchy’s theorem and its consequences. 7. To define Laurent series, the isolated singularities, to expand a function in a Laurent series. To determine the type of the singularity of a given function. 8. To define the concept of the residue. To formulate and prove theorems that relate to its application to the calculation of integrals of complex functions. 9. To define the conformal mappings. To formulate and prove theorems that relate to the properties of conformal mappings. 10. To define the bilinear mapping. To indicate and prove their properties.