Department of Mathematics
Stony Brook University
office: Math Tower 5-111
phone: (631) 632-8287
e-mail: leon.takhtajan@stonybrook.edu
| Dates | Sections
covered and assigned reading |
Homework |
| Feb 4 & Feb 9 |
Complex numbers. Complex
differentials and holomorphic functions, Cauchy-Riemann
equations. Elementary theory of power series. Ch.1 and Ch.2, §§1.1-1.4 and §§2.1-2.4. |
|
| Feb 11 & Feb 16 |
The exponential and
trigonometric functions. The logarithm. Holomorphic functions as mappings. Conformality. Ch.2, § 2.5 and §§3.1-3.4, Ch.3, §§ 2.1-2.4. |
| Feb 18 & Feb 23 |
Fractional linear
transformations. Elementary conformal mappings. Complex integration. Ch.3, §§3.1-3.5 and §§4.1-4.2, notes on Zhukovsky function and Ch. 4, §§1.1-1.3. |
Extra Credit due: Feb 25 |
| Feb 25 & Mar 2 |
Complex integration. Goursat
and Cauchy theorems. Cauchy integral formula and power
series. Cauchy’s inequalities and consequences: Liouville's theorem
and fundamental theorem of algebra. Ch.4, §§1.1-1.5 and §2.3. Presentation in class will differ from Ahlfors; I will simplify exposition in the textbook. |
|
| Mar 4 & Mar 9 |
Morera's theorem and
Weierstrass theorem on uniform limits of holomorphic
functions. Local properties of holomorphic functions. Index of a
point with respect to a closed curve and strong form of Cauchy
integral formula. Open
mapping theorem and maximum modulus principle.
Ch.4, §§2.1-2.2, §§3.1-3.4 and Ch.5, §1.1. Presentation in class will differ from Ahlfors. I will simplify some details in the textbook. |
|
| Mar 11 & Mar 16 |
Maximum modulus principle and
Schwarz lemma and conformal autmorphisms of the Riemann sphere,
the complex plane and the unit disk.
Ch.4, §3.4. |
|
| Mar 23 & Mar 25 |
The general form of Cauchy
theorem. The residue theorem, the
residue at infinity and sum of residues theorem.
Ch.4, §§4.1-4.6 and §5.1. Presentation in class will differ from Ahlfors, we will prove a homotopy version of Cauchy theorem. |
|
| Mar 30 & Apr. 1 |
The residue theorem cont. The
argument principle and Rouche's theorem. Hurwitz theorem as
application of Rouche theorem. Different proofs of
Jensen's formula.
Shabat notes §3.3, Ahlfors Ch.4, §§5.1-5.2 and Ch.5, §3.1, and notes on the Blackboard. |
|
| Apr 6 & Apr 8 |
Evaluation of definite
integrals. Jordan lemma. Introduction to Euler gamma function and
Riemann zeta function.
Shabat notes §3.3, Ahlfors Ch.4, §5.3 and Ch.5, §1.1. Presentation in class will differ from Ahlfors, we will introduce the gamma function by the Mellin transform. |
|
| Apr 13 & Apr 15 |
The Laurent series. Partial
fractions and factorization, Mittag-Leffler and Weierstrass theorems.
Ch.5, §§1.1-1.3 and §§2.1-2.3. |
|
| Apr 20 & Apr 22 |
Order and genus of entire
function. Hadamard and Picard theorems (without a proof). Normal
families, Montel theorem, the Riemann mapping theorem.
Ch.5, §§3.2 and §§5.1-5.5, Ch.6, §1.1. & Ch.8, §3.1. Our exposition will be somewhat different from Ahlfors: we will do only what is necessary for the proof of the Riemann mapping theorem. |
|
| Apr 27 & Apr 29 |
The Riemann mapping theorem:
Riemann's original proof. Boundary behavior. Schwarz reflection principle. Modular λ-function
and Picard theorem.
Ch.6, §§1.2.-1.4. |
| May 4 & May 6 |
Harmonic functions. Functions with mean-value
property. Harnack's principle. The Dirichlet problem and subharmonic functions.
Ch.4, §§6.1.-6.5, Ch.6, §§3.1-3.2 and §§4.1-4.2. |
|