Complex functions, 5 cr

Core content

• Complex 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 knowledge

• Visualization of complex functions
• Applying residues to calculate real integrals.

Specialist knowledge

• Liouville’s theorem and the fundamental theorem of algebra as a consequence of Cauchy’s integral formula
• Applications: Harmonic functions and/or conformal mappings