Title of the course: A crash course of complex analysis
Instructor’s Name: Assoc. Prof. Uğur GÜL
Institution: Hacettepe Ü.
Dates: 24 August-04 September 2020
Prerequisites: A solid background of Calculus and Advanced Calculus(Familiarity with line integrals and Green’s theorem will be especially needed)
Level: Graduate, advanced undergraduate
Abstract:
Week 1:
Lecture 1: Complex numbers and complex number system, polar representation, Euler’s formula, absolute value and argument of a complex number, complex conjugate, multiplication of complex numbers, roots of unity
Lecture 2: Topology of the complex plane, open and closed sets, accumulation points, closure and boundary of a subset of the complex plane, Riemann sphere and point at infinity, Steographic projection
Lecture 3: Functions on the complex plane, continuous functions, differentiable functions, del and delbar derivatives, holomorphic functions, Cauchy-Riemann equations
Lecture 4: Piecewise continuous curves in the complex plane, complex line integrals, Anti-derivatives and fundamental theorem of integral calculus for complex line integrals, Cauchy-Goursat theorem, Cauchy integral theorem
Lecture 5: Consequences of Cauchy integral theorem: Cauchy integral formula, Gauss mean value theorem, Taylor’s theorem for complex functions, Cauchy integral formula via Green’s theorem and generalizations: Cauchy-Fantappié integral formula, Cauchy estimates on derivatives, Liouville’s theorem
Week 2:
Lecture 1: Power series and convergence of power series, Disc and radius of convergence of power series, Cauchy-Hadamard formula, Analytic functions, Analytic functions are holomorphic
Lecture 2: Some elementary functions: polynomials, fundamental theorem of algebra, exponential function, trigonometric functions, complex logarithm function Log, branch cuts.
Lecture 3: Isolated singularities of analytic functions: meromorphic functions, singularity types: Riemann removable singularity theorem, Casorati-Weierstrass theorem, Laurent series, Residue theorem.
Lecture 4: Sequences of holomorphic functions, convergence of holomorphic function sequences: uniform convergence on compact subsets, Weierstrass’ theorem on uniform convergence on compact subsets, Hurwitz theorem, normal families: Montel’s theorem.
Lecture 5: Infinite products, convergence criteria of infinite products, determination of holomorphic functios with prescribed zeros: Weierstrass theorem, determination of meromorphic functions with prescribed poles: Mittag-Leffler theorem.
Language: ENG
Textbooks: L. V. Ahlfors, “Complex Analysis”
J. Conway “Functions of one complex variable”
T. W. Gamelin “Complex Analysis”