Learning advanced complex calculus and it’s use in various applications: conformal mappings, integral transforms, orthogonal polynomials and special functions.
- operate with elementary functions in complex domain
- define and examine analytic property of functions
- apply the technique of conformal mappings in analyze of simple models
- use the technique of power series expansion in the calculus with elementary functions
- classify zeroes and singular points of analytic functions
- apply the technique of residuum, especially in the inverse integral transformation
- use the properties of gamma and beta 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||Threshold||Percent of Grade|
|Mid Term Exam: Written||0 %||50 %||0 %|
|Final Exam: Written||0 %||50 %|
|Exam: Written||0 %||100 %|
Week by Week Schedule
- Complex plane. Regions. Contours.
- Functions of complex variables. Multiple-valued functions. The power and root functions.
- Exponential and logarithmic function. Trigonometric and other elementary functions.
- Differentiability. Analytic functions. Cauchy-Riemann conditions. Harmonic functions.
- Bilinear transformation (Möbius transform) and examples.
- Conformal mappings. Examples of region mappings.
- Integral of functions of complex variable. Independence of path. Cauchy theorem. Cauchy integral formula. Applications.
- Taylor series. Zeroes of analytic functions.
- Laurent series. Singular points and poles of analytic functions.
- Residue calculus and residue theorem.
- Applications of residue. Inverse Laplace transform.
- Special functions. Gamma function.
- Beta function. Applications.