Complex Analysis
Instructor:Guanghua Ji
Textbook:Elias M. Stein, Rami Shakarchi, Complex Analysis, Princeton Univ Press.
Grading: Be based on homework(20%), midterm exam(30%) and final exam(50%).
Schedule:TUE 9-10; FRI 3-4, 119.
References:
1, 余家荣等, 复变函数专题选讲, 高等教育出版社, 2012.
2, Joseph Bak, Donald J. Newman, Complex Analysis, Springer-Verlag, New York, 2010.
3, Sheng Gong, Youhong Gong, Concise Complex Analysis, World Scientific Press, 2007.
4, 扈培础， 复变函数教程， 科学出版社， 2008.
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I, Cauchy Integral Theory
Chapter 1. Differential Calculus in the Complex Plane
9/8,§1.1, Complex numbers and complex functions:Basic properties: complex numbers, complex plane.§1.2, Complex functions:Continuous functions, holomorphic function HW: p.24, 2,3,7.
9/10, Cauchy-Riemann equation.§1.3, Power series: Radius of convergence, analytic.§1.4, Integration along curves.HW: p.24, 8,10,13.
Chapter 2. Cauchy’s Theorem and Its Applications
9/18,§2.1, Goursat's Theorem:Integration along curves; Goursat's theorem: triangle case. HW: p.24, 19,20,23,25.
9/22,§2.2, Cauchy's Theorem in a dics and Cauchy's intergral formulas:HW: p64, 5,6.
9/23,§2.3, Cauchy's intergral formulas and app:Cauchy inequalities, Taylor series, Liouville's theorem, the fundamental thm of algebra, zeros of holomorphic functions are isolated. HW: p64, 7,8,9.
9/29,§2.4, Further Applications:Morera's thm, sequences of holo functions, holo fun defined in term of integrals, Schwarz reflection principle. HW: p64, 11,13,15.
II, Weierstrass Series Theory
Chapter 3. Meromorphic Functions and the Logarithm
9/30,§3.1,Laurent Series and isolated singularity:Laurent series; Isolated singularities: removable singularities, pole, essential singularities. HW: 13,14,15.
10/8,§3.2,The Residue formula and Appliacations:HW
10/9,§3.3, The argument principle and applications:I, Argument principle. Rouche's theorem, open mapping theorem, Maximum modulus principle. II, Mean value Property. HW: p106, 16,17,19.
10/13,§3.4, Generized Cauchy's theorem and The complex logarithm:I, Homotopies and generized Cauchy's theorem; II, Analytic branch, power functions. HW:
10/14,§3.5,The Evaluation of integrals:I, Complex integrals. II, Real integrals. III, Evaluation and esimation of sums. HW: p103, 2,3,4,7,9,11.Homework Supplements
10/20, Recitation.
Chapter 4. The Fourier Transform
10/21,§4.1, Poisson formula summation:Holomorphic fourier transform HW: p127,1,7.
10/27,§4.2, Paley-Wiener Theorem:HW: p128,8.
Chapter 5. Entire Functions
10/28,§5.1, Functions of finite order:I, Jensen's formula. II, entire functions of finite order. HW: p153, 2,4.
11/3,§5.2,Weierstrass infinite products: I, Infinite products. II, Weierstrass infinite product. III, Hadamard's factorization theorem. HW: 7,10,13,14.
IV, Applications
Chapter 6.Applications to Number Theory
11/4,§6.1, The Gamma function:I, Analytical continuation and functional equation. II, Euler-Gauss formula, Weierstrass formula. III, Euler's reflection formula, and Legendre formula. HW: p174, 1,3,6.
11/10,§6.2, The Riemann zeta function:I, Definition and Euler product, II, Analytical continuation and functional equation. HW: p178, 15. p180, 2.
11/11,§6.3, Zeros of the Riemann zeta function:I, Trivial zeros. II, critical strip. III, Zero-free region. IV, Estimates for 1/\zeta(s). HW, p200, 3,6,7.
11/17,§6.4, Prime number theorem:I, The Chebyshev functions. II, Proof of PNT. HW.10,12.
11/18,§6.5, Elliptic functions:I, Elliptic functions II,The Weierstrass elliptic function. III, The Jacobi elliptic functions. HW. p278, 1,3.
11/24,§6.6, Eisenstein series:I, Eisenstein Series. II, Fouier expansion. HW. p278, 7.
11/25,§6.7, Modular functions: I, Full modular group; II, Classical modular functions. III, The Linear space of modualr forms.
12/1,§6.8, Riemann surfaces: I, Multiple valued functions; II, The general definition: complex manifolds; III, Examples.
III, Riemann Geometry Theory
Chapter 7. Conformal Mappings
12/2,§6.1, Conformal mapping connected with elementary functions:I, Conformal equivalenceII, Elementary transformations. III, fractionla linear transformations. HW.p248, 1,4,5.
12/8,§6.2, The Schwarz lemma and App: I,The Schwarz lemma. II,Automorphisms of the disc and the upper half plane. III, The Schwarz-Pick Theorem. HW.p249, 10,11,12,13,14.
12/9,§6.3,The Riemann mapping theorem:I, Arzela-Ascoli theorem. II,Montel's theorem. III,Hurwitz's theorem. HW: p252, 16,17..
12/15, VI, TheRiemann mapping theorem.§6.4,Harmonic functions and Dirichlet Problems: I, Fundamental properties; Def; mean value theorem; maximum modulus theorem.
12/16,II, Poisson formula III, Dirichlet problem: subharmonic. HW:
12/22,Recitation.
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Final Exam
Homework Supplements
Yau College Student Maths Contests
Online resources
