Function of one complex variable

Prerequisites: 

Fundamental Algebra and Analysis II

Course description:

This course is designed for second year students who intend to pursue studies in fundamental mathematics or applied mathematics major. The teaching objective is to enable students to master the fundamental knowledge and language of complex analysis, in order to prepare them for in-depth study of modern mathematics. This course will employ a variety of teaching methods, including lectures, class discussions, problem-solving sessions, and independent reading.

Learning Objectives:

Upon completion of this course, students will be able to:

Understand the structure of complex numbers and perform algebraic and geometric operations in the complex plane.

Define and analyze holomorphic functions, apply the Cauchy-Riemann equations, and interpret their geometric meaning.

Utilize Cauchy's integral theorem and formula to evaluate integrals and represent holomorphic functions via power series.

Understand basic properties of holomorphic functions

Classify isolated singularities and compute residues to evaluate real and complex integrals. Understand and apply the residue theorem, argument principle, and Rouché's theorem. Explore conformal mappings and basic properties of Möbius transformations.

Gain introductory exposure to special functions (Gamma, Zeta)

Understand the proof of key theorems such as the Riemann mapping theorem and prime number theorem.

Detailed Topics Covered:

1. Complex numbers: algebraic and geometric viewpoints, polar form, roots of unity.

2. Topology and metric on the complex plane: open sets, compactness, connectedness, homotopy.

3. Holomorphic functions: definition, Cauchy–Riemann equation, differentiation rules, geometric meaning of holomorphicity.

4. Power series: radius of convergence, Taylor expansion, power seris as holomorphic functions, power seris expansion for e^z,sinz,cosz,log(1+z)

5. Complex integration: rectifiable curves, line integrals, Cauchy's theorem and formula, winding number.

6. Basic properties of holomorphic functions: Holomorphic function is locally a power series, isolated zeros, Cauchy's estimate, Liouville's theorem, Fundamental theorem of algebra,local uniform limit of holomorphic functions is holomorphic, Morera's theorem,existence of primitive function, local normal form,open mapping theorem, injective implies biholomorphic, maximal modules principle.

7. General form of Cauchy's theorem and formula for cycles homologues to zero.

8. Singularities and residues: Laurent series,classification, residue theorem, applications to integration.

9. Meromorphic functions: Riemann sphere as first example of Riemann surface,argument principle, Rouché's theorem, rational maps.

10. Möbius transformations, cross-ratio,symmetric principle.

11. Schwarz lemma, hyperbolic metrics, Schwarz-Pick theorem.

12. Normal families, Ascoli-Arzela theorem, Hurwitz theorem and the proof of the Riemann mapping theorem

13. Infinite products,genus of entire functions, Weierstrass factorization.

14. Special functions: Gamma function, Riemann zeta function,non-trivial zeros of Riemann zeta function.

15. The proof of the prime number theorem.