Complex Functions (5cr)
Course unit code: C-10122-MATH--APP--440
General information
- Credits
- 5 cr
- Institution
- University of Tampere
Objectives
On the course students learn the basics results related to differentiation and integration of complex functions starting from the definition of complex numbers. After passing the course the student: is familiar with the complex numbers and their basic operations and can interpret them geometricallyknows the basic functions and their properties and can solve complex equations involving themcan decide when a function is analytic, e.g., by using the Cauchy-Riemann equations and knows the main features of analytic functionsknows the definition of complex integrals and the related basic results, and can calculate complex integrals using themcan find the Taylor’s or Laurent's series of a given function and can study their convergence knows what a singularity of a function is and can deduce its type can make logical conclusions, e.g., is able to make mathematical proofs concerning complex functions and explain why certain results may be used
Content
Core contentComplex numbers, basic operations and their geometric interpretation. Elementary complex functions and their properties.Continuity, differentiability and analyticity of complex functions. Cauchy-Riemann equations.Complex Integral and the related basic results: The fundamental theorem of analysis, Cauchy-Goursat theorem, Cauchy's integral theorem, Cauchy's integral formulas, and deformation theorem.Taylor's and Laurent's series and their applications: Singularities and Residue calculus.Complementary knowledgeVisualization of complex functionsApplying residues to calculate real integrals.Specialist knowledgeLiouville’s theorem and the fundamental theorem of algebra as a consequence of Cauchy’s integral formulaApplications: Harmonic functions and/or conformal mappings
Prerequisites
Prerequisite information consists of Differential and Integral Calculus, or any combination of courses with corresponding contents. In particular, basic knowledge of differentiation and integration of real functions is required. Knowledge on the basic contents of Multivariate calculus, e.g., on visualization of two-variable real functions and partial differentiation, is recommended.
Further information
Will be lectured only in the academic year 2025-26.