Learning Outcomes
1. Perform calculations with complex numbers and use elementary complex functions as a tool of geometric
transformations.
2. Use complex integration as a tool for qualitative study of two dimensional vector fields (circulation-
flux, complex potential).
3. Use complex analysis techniques as a tool in signal processing (calculating Fourier, Laplace and zeta
transforms).
4. Use conformal mappings for solving problems of electromagnetism (calculating the electric field in a region, given boundary conditions) and for solving PDE’s.
Course Content (Syllabus)
Algebra of complex numbers, polar form, roots of unity. Functions of a complex variable (exponential, logarithmic, trigonometric, hyperbolic, linear functions and inverse functions). Limit, continuity and complex derivative. Holomorhic (analytic) functions and Cauchy-Riemann equations. Complex integration. Cauchy theorem and Cauchy integral formulas. Applications (Liouville theorem, Cauchy inequality, max/min modulus principle). Series of complex numbers. Taylor and Laurent series. Removable singularities, poles and essential singularities. Residuals and applications. Conformal mappings. Ηarmonic and conjugate harmonic functions. Dirichlet's problem for the disk and half plane.
Keywords
Analytic functions, Cauchy theorem, residuals, harmonic functions