Learning advanced complex calculus and it’s use in various applications: conformal mappings, integral transforms, orthogonal polynomials and special functions.
- perform basic operations with complex numbers in various representations
- understand the concept of elemenary function and their interconnection
- apply the technique of conformal mappings in analyze of simple models
- use the technique of power series expansion in the calculus with elementary functions
- apply the technique of residuum, especially in the inverse integral transformation
- understand the concept of orthogonal systems and expansion into series of orthogonal polynomials
- use the properties of gamma function in various situations
Forms of Teaching
Lectures are organized through two cycles. The first cycle consists of 7 weeks of classes and mid-term exam, a second cycle of 6 weeks of classes and final exam. Classes are conducted through a total of 15 weeks with a weekly load of 3 hours.Exams
Mid-term exam in the 8th week of classes and final exam in the 15th week of classes.Consultations
Consultations are held one hour weekly according to arrangement with students.
|Type||Threshold||Percent of Grade||Comment:||Percent of Grade|
|Quizzes||0 %||20 %||0 %||20 %|
|Mid Term Exam: Written||0 %||40 %||0 %|
|Final Exam: Written||0 %||40 %|
|Exam: Written||0 %||80 %|
If student hasn't attended quizzes or wants to improve obtained results, he can take a written exam for 100% of grade.
Week by Week Schedule
- Complex plane. Regions. Contours. Sequences and series, Convergence.
- Functions of complex variables. Differentiability. Analytic functions. Cauchy-Riemann conditions. Harmonic functions.
- Conformal mappings. Multiple-valued functions. The power and root functions. Branch point.
- Exponential and logarithmic function. Trigonometric and other elementary functions.
- Bilinear transformation. Conform Mappings. Examples of region mappings.
- Integral of functions of complex variable. Independence of path.
- Cauchy theorem. Cauchy integral formulas. Applications.
- Taylor series. Zeroes of analytic functions
- Laurent series. Singular points and poles of analytic functions.
- Residue theorem and applications. Inverse Laplace transform. Applications.
- Special functions. Gamma and beta functions.
- Bernoulli polynomials. Generating functions. Orthogonal polynomials and corresponding differential equations.
- Legendre polynomials. Hermite and Čebišev polynomials. Applications.