**Complex Analysis**

**Instructor**: Guanghua Ji

**Grading:** Be based on homework(20%), midterm exam(30%) and final exam(50%).

**Schedule: **TUE 9-10; FRI 3-4; 315.

**References**:

1, Elias M. Stein, Rami Shakarchi, Complex Analysis, Princeton Univ Press.

2, Sheng Gong, Youhong Gong, Concise Complex Analysis, World Scientific Press, 2007.

3, Serge Lang, Complex Analysis, fourth edition. Springer-Verlag.

4, 方企勤， 复变函数教程， 北京大学出版社， 2017.

5, 扈培础, 复变函数教程, 科学出版社, 2008.

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**Chapter 1. Differential Calculus**

§1.1, Complex numbers and the complex plane:I, complex numbers. II, Topology of complex plane.

§1.2, Complex functions:I, Holomorphic function. II, Cauchy-Riemann equation. HW: p.24, 2,3,7.

§1.3,Integration along curves:I, Parametrized curves. II, Integral along curves. III, Primitive functions. HW: p.24, 8,10,13.

§1.4, Power series:Cauchy criterion, Weierstrass M-test, Abel theorem. HW:

§1.5, Elementary functions:I, exponential functions. II, logarithmic function. III, Trigonometric functions and their inverses. HW:

**Chapter 2. Cauchy’s Integral Theorey**

§2.1, Goursat's Theorem and Cauchy's Theorem in a dics:I, Goursat's theorem: triangle case. II, Cauchy's thm in a disc. HW: p.24, 19,20,23,25.

§2.2, Homotopic version of Cauchy's theorem:I, Homotopies. II, Cauchy's thm on simply connected domains. III, General forms. IV, the complex logarithm: analytic branch. HW: p64, 1,2,3,5,6.

§2.3, Cauchy's intergral formulas and applications:Cauchy inequalities, Taylor series, Liouville's theorem, the fundamental thm of algebra, zeros of holomorphic functions are isolated. HW: p64, 7,8,9.

§2.4,Further Applications:I, Morera's theorem. II, Weierastrass theroem: sequences of holomorphic functions. III, holomorphic functions defined in term of integrals. IV, Schwarz reflection principle.

§2.5, Maximum modulus principle and Schwartz LemmmaI, Meav-valued property. II, Maximum modulus principle. III, Schwartz lemma. VI, The Schwarz-Pick Theorem. HW: p66, 11,13,15.

**Chapter 3. Weierstrass Series Theory**

§3.1,Laurent Series and isolated singularity:I, Laurent series. II,Isolated singularities: removable singularities, pole, essential singularities, Riemann’s Principle, Casorati-Weierstrass Theorem. III, Isolated singularity at infity. HW: p105,, 13,14,15.

§3.2,Entire functions and meromorphic functions:I, entire functions: infinity point. II, meromorphic functions: definition, meromorphic functions on the extended complex plane, partial fractions decomposition. HW: p106, 16,17.

§3.3,The Residue formula, argument principle and applications:I, the residue formula. II, Argument principle. III, Rouche's theorem. VI, open mapping theorem. V,Hurwitz's theorem. HW: p107, 19,20.

§3.4 The Evaluation of definite integrals:I, trigonometric integrals. II, rational functions and fourier transforms: two lemmas. III, Integral along a branch cut. Mellin transform. HW: p103, 2,3,4,5,6,7,8,9,10,11.**Homework Supplements**

§3.5, Functions of finite order and Infinite Products:I, Jensen's formula. II, entire functions of finite order. III, Infinite products. HW: p153, 2,3,4.

§3.6, Weierstrass infinite products: I, Weierstrass infinite product. II,Hadamard's factorizationtheorem. III,Mittag-Leffler Theorem. HW: 7,10,12,13,14.

**Chapter 4. Conformal Mappings and Riemann Mapping Theorem**

§4.1, Conformal mapping connected with elementary functions:I, Conformal equivalenceII, Elementary transformations. III, fractional linear transformations. HW.p248, 1,4,5.

§4.2, Groups of HolomorphicAutomorphisms: I,Automorphisms of the complex plane. II, Automorphisms of the complex plane the disc. III, Automorphisms of the upper half plane. HW.p249, 10,11,12,13,14.

§4.3,Normal family and Montel's Theorem:I,Arzela-Ascoli theorem. II, Normal family. III,Montel's theorem. HW: p252, 16,17.

§4.4,The Riemann mapping theoremThe Riemann mapping theorem.

**Chapter 5.Applications**

§5.1,The Gamma function:I, Analytical continuation and functional equation. II, Euler-Gauss formula, Weierstrass formula. III, Euler's reflection formula and Legendre formula. IV, Stirling's asymptotic formula. HW: p174, 1,3,6.

§5.2, Fourier transfrom and Poisson formula summation:I The Schwartz class. II, Fourier inversion formula. III, Poisson summation formula. VI, Theta function. HW: p127, 2,3,6.

§5.3, The Riemann zeta function:I, Convergence. II, Euler product. III, Analytical continuation. IV, Functional equation. V, Distribution of zeros: Trivial zeros, infinitly many zeros in the critical strip, RH; zero-free region. HW, p178,15; p180, 2; p200, 3,6,7.

§5.4, Prime number theorem:I, Abel's lemma. II, The Chebyshev functions: equivalent statements. III, Newman’s Short Proof of the Prime Number Theorem. HW.10,12.

§5.5, Elliptic functions:I, Elliptic functions II,The Weierstrass elliptic function. III, Eisenstein series. HW. p278, 1,2,3,7.

§5.6,Harmonic functions and Dirichlet Problems: I, Fundamental properties; Definition; mean value theorem; maximum modulus theorem. II, Poisson Formulae. III, the Dirichlet Problem

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**Final Exam**

**Homework Supplements**

**Yau College Student Maths Contests**

**Online resources**