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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.


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.


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:



Final Exam

Homework Supplements

Yau College Student Maths Contests

Online resources

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