Course information

Complex Numbers: Conjugate, modulus, and polar form of a complex number. Sequences and series of complex numbers. Complex functions, limits, and continuity. Exponential, logarithmic, and trigonometric complex functions. Differentiable complex functions: Cauchy—Riemann equations. Holomorphic functions and basic properties. Complex line integral and basic properties: Cauchy—Goursat theorem. Deformation principle. Cauchy’s Integral Formulas and consequences: Maximum Modulus Principle, Fundamental Theorem of Algebra, Liouville’s theorem. Power series and radius of convergence: Taylor’s theorem and Taylor’s expansions of basic entire functions. Identity theorem. Laurent’s theorem and Laurent’s expansions. Isolated singular points of a complex function: Poles, removable and essential singular points. Calculus of residues: Applications to the calculation of trigonometric and improper real integrals. Conformal mappings: Möbius transformations and applications.