复分析-1
Complex analysis
Analytic function, Cauchy’s Integral Formula and Residues, Power Series Expansions, Entire Function, Normal Families, The Riemann Mapping Theorem, Harmonic Function, The Dirichlet Problem Simply Periodic Function and Elliptic Functions, The Weierstrass Theory Analytic Continuation, Algebraic Functions, Picard’s Theorem
1，复数，复数域、复平面、平面点集，区域与曲线，复平面上的直线与半平面、球极投影，Riemann球，扩充复平面及其球面表示、幂级数。
2，单复变量函数，极限与连续，复变量函数的可微性，幂级数、解析函数、Cauchy-Riemann条件，Cauchy-Riemann方程、解析函数、全纯函数。共形映射、分式线性变换、Mobius变换、导数的几何意义，共形映射、对称原理。Riemann定理。
4。初等函数，他们的性质，初等函数与共形映射的关系（整分式线性变换与分式线性变换，将上半平面的边界映射为平面上的圆周的分式线性变换，指数与对数曲线，任意次数幂函数，Riemann曲面的概念，指数与对数函数的Riemann曲面，Zhukovsky函数，三角与双曲函数）
5。复变量函数的积分及其基本性质，复变量函数的积分与第一型和第二型曲线积分的联系，化为实变量函数的积分，原函数，Newton-Leibnitz公式，积分号下取极限。
3， 有界变差函数、Riemann-Stieltjes积分。
4， Cauchy估计公式、解析函数的幂级数表示、整函数、解析函数的零点、Liouville定理、代数基本定理、最大模定理、闭曲线的指标。
5， Cauchy定理、Cauchy积分定理，Cauchy积分公式、Cauchy型积分。
无穷可微解析函数，导数的Cauchy公式，Morera定理、零点的计算、开映射定理。
8。数列与解析函数的级数，Weierstrass定理，函数空间，区域上的解析函数。
9。幂级数，解析函数的幂级数展开，展开的唯一性，系数的Cauchy公式与不等式，Liouville定理，幂级数的应用。
10。解析函数的唯一性定理，解析函数的零点，零点的次数。
6， Goursat定理、奇点的分类、可移奇点定理，单值函数的孤立奇点及其分类，
Laurent级数，Laurent级数展开、它的收敛域，解析函数的Laurent级数展开，展开的唯一性，系数的Cauchy公式与不等式。Casorati-Weierstrass定理。Sokhotskogo-Weierstrass定理，Picard定理，孤立奇点作为特殊的无穷远点。
7，留数，留数定理、留数的计算公式，留数的Cauchy定理。
14。运用留数计算积分，Jordan引理。
对数留数，辐角原理、Rouche定理、解析函数所作的余缺的映射，最大模原理。
8， Schwarz引理、Hadamard三圆定理、Phragmen-Lindeloff定理、Arzela-Ascoli定理。
9， 解析函数空间、Hurwitz定理、单叶函数的收敛级数。Montel定理、亚纯函数空间、Riemann映射定理。
10， Weierstrass因式分解定理、正弦函数的因式分解、Runge定理。
18。单叶性的局部判据法，解析函数的逆，边界对应原理，分式线性映射的基本性质。
19。解析开拓，完全解析函数，完全解析函数的Riemann曲面与奇点，单值定理。
20。沿有界区域的解析开拓，对称原理及其在共形映射中的应用。
11，整函数，整函数的阶和型，Weierstrass乘积，亚纯函数，扩充平面上的亚纯函数，单连通性、Mittag-Leffler定理、Schwarz反演原理。
12， 函数芽、沿道路的解析开拓、完全解析函数、单值性定理、调和函数、最大值原理、最小值原理、Poisson核、Harnark不等式、Harnark定理。
13， 次调和函数与上调和函数、Dirichlet问题、Green函数。
14， Jensen公式、Poisson-Jensen公式、Hadamard因式分解定理。
复分析-2
1， Bloch定理、Picard小定理、Schottky定理、Montel-Caratheodory定理、Picard大定理、共形映射在流体力学上的应用。
2，多角形的共形映射，Pompeiu公式、Schwarz-Christoffel公式。
3， Gamma函数、亚纯函数的Nevanlinna定理。
Laplace变换、渐进级数、渐进展开、Riemann-Zeta函数。
4， Green公式、椭圆函数与双周期性、Liouville定理、因子群、
椭圆函数，Weierstrass椭圆函数。Jacobi椭圆函数，Riemann-Zeta函数，用Riemann-Zeta函数表示Jacobi椭圆函数，Jacobi椭圆函数的加法公式。
5， 椭圆函数域、椭圆积分。
6， 加性定理、椭圆函数论在椭圆积分上的应用。
7， Abel定理、椭圆模群。
8， 模函数、Picard小定理。Eisentein级数。Montel定理。
9， 模群及其基本域。
10， 模形式的代数、Theta函数的Jacobi变换公式。
24。Riemann存在定理，共形映射的唯一性条件。
11， 同余群、同余群的模形式、单连通流形上的函数的整体连续。
12， 曲面的定义、Riemann曲面、Riemann曲面上的Rieman度量、Laplace-Beltrami算子、Schwarz-Pick定理、双曲度量、测地线。
13， 双曲同构的离散群、基本多边形、Riemann曲面上的Gauss-Bonnet公式、Riemann-Hurwitz公式。
26。调和函数，调和函数与解析函数的联系，无穷可微调和函数，平均值定理，唯一性定理，最大值与最小值原理，Liouville与Harnak定理，Poisson与Schwarz积分，调和函数的级数展开及其与三角级数的联系。
27。拟共形映射，Dirichlet问题及共形映射在求解Dirichlet问题中的应用。28。调和函数与解析函数在流体力学中的应用。
Complex Analysis & Riemann Surface
- Valerian Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable
- K. Kodaira, Complex Analysis
- Rudin, Real and complex analysis
- 龚升，简明复分析
1，A.I.Markushevich，解析函数论简明教程，1987。
2，I.I.Privalov，复变函数引论，1984。
3，B.V.Shabat，复分析导论，第一卷，1986。
4，M.A.Lavrentyev、B.V.Shabat，复变函数论方法，1987。
5，M.A.Evgrafov、Yu.B.Sidorov、M.V.Fedoryuk、M.I.Shabunin、K.A.Bezhanov，解析函数论习题集，1972。
6，T.A.Leontyeva、Z.S.Panferov、B.C.Serov，复变函数论习题集，莫斯科大学讲义，1992。
7，E.P.Dolzhenko、S.N.Nikolaeva，复变函数论学习指导书，莫斯科大学讲义，1988。
补充参考书目：
1，A.V.Bitsadze，单复变函数论基础，1972。
2，A.I.Markushevich，解析函数论，1967。
3，V.I.Smirnov，高等数学教程，1972。
4，A.G.Sveshnikov、A.N.Tikhonov，复变函数论，1974。
5，Y.V.Sidorov、M.V.Fedoryuk、M.I.Shabunin，复变函数论讲义，1976。
7，L.I.Volkovysk、G.L.Lunts、I.G.Aramanovich，复变函数论习题集，1975。
Titchmarch “函数论”
戈鲁辛 “复变函数几何理论”
Conway, “Functions of One Complex Variable”
Hormander “An Intro to Complex Analysis in Several Variables”
H.Cartan “解析函数论引论”《解析函数论初步》科学出版社
Beardon, “Complex Analysis”
R.Remmert “Complex Analysis”(GTM,reading in mathematics)
Steven G. Krantz：Function Theory of Several Complex Variables
Steven G. Krantz：Complex Analysis: The Geometric Viewpoint
Lang, Complex analysis：
Elias M. Stein：Complex Analysis
方企勤，《复变函数教程》，北大版
史济怀，《多复变函数论基础》，高教版
张南岳，《复变函数论选讲》，北大版
任尧福《应用复分析》复旦大学出版社
学复变函数中“古典分析”之外的理论，比如共形映射，
范莉莉,何成奇 “复变函数论”
庄(欽/圻)泰,何育瓒等 “复变函数论(专题?)选讲”
余家荣《复变函数》高等教育出版社
《复变函数》钟玉泉
J.-P. Serre, “A course of Arithmetics”
O.Forster：Lectures on Riemann Surfaces
Jost：Compact riemann surfaces
Narasimhan：Compact riemann surfaces
Lang：Riemann surfaces ,
Hershel M. Farkas：Riemann Surfaces
143《复变函数》大连理工数学系组编
【习题集与辅导书】
145《高等数学例题与习题集.三,复变函数》博亚尔丘克编著
【提高】
科大严镇军也有一本《复变函数》
Complex analysis -1
Complex analysis
Analytic function, Cauchy’s Integral Formula and Residues, Power Series Expansions , Entire Function, Normal Families, The Riemann Mapping Theorem , Harmonic Function, The Dirichlet Problem Simply Periodic Function and Elliptic Functions , The Weierstrass Theory Analytic Continuation, Algebraic Functions, Picard’s Theorem
1 , Complex numbers, Plural fields, the complex plane, Planar point set, and curves, Line in the complex plane and the half plane, The stereographic projection, Riemann Ball, extended complex plane Spherical representation, and power series.
2 , Functions of a complex variable, limit and continuity, and differentiability of functions of a complex variable, Power series, analytical functions, Cauchy-Riemann Conditions, Cauchy-Riemann Equations, analytic functions, Holomorphic functions. Conformal mapping, the fractional linear transform,Mobius transform, the geometrical meaning of the derivative, Conformal mapping, symmetry principles. Riemann theorem.
4 。 Elementary functions and their properties, elementary functions and Conformal mapping relationships (fractional linear transforms and fractional linear transformations, maps the upper half-plane boundary for the circle in the plane of fractional linear transformations, exponential and logarithmic curve, any number of times power function,Riemann surfaces, the concept of exponential and logarithmic functions, Riemann surfaces, Zhukovskyfunctions, trigonometric and hyperbolic functions)
5 。 Integral and its basic properties of functions of a complex variable, functions of a complex variable of integration and the first and second line integrals of the type of contact, as the real variable function of integral, primitive function,Newton-Leibnitz equation, integral sign under the limit.
3 , Bounded variation function,Riemann-Stieltjes points.
4 , Cauchy estimation formula, the power series representation of analytic functions, the whole functions, zeros of analytic functions, andLiouville theorem, the fundamental theorem of algebra, closed curve of maximum modulus theorem, indicator.
5 , Cauchy theorem andCauchy integral theorem andCauchy integral formula, theCauchy type integrals.
Analysis of infinitely differentiable functions, the derivative of Cauchy Formula Morera Theorem, zero-point calculation, the open mapping theorem.
8 。 Sequences and series of analytic functions,Weierstrass theorem, function spaces, analytic functions on the region.
9 。 Power series, power series expansion of analytical functions, uniqueness, the coefficients of the Cauchy equation and inequality,Liouville theorem, the application of power series.
10 。 The uniqueness theorem for analytic functions, of zeros of analytic functions, zero times.
6 , Goursat theorem, the singularity of the classification, removable Singularity theorem, isolated singularities and classification of single-valued functions,
Laurent Series, Laurent Series expansion, and Its region of convergence and analytic functions Laurent Series expansions, expand the uniqueness coefficients Cauchy Equations and inequalities. Casorati-Weierstrass theorem. Sokhotskogo-Weierstrass theorem,Picard theorem, isolated singular point at infinity as a special.
7 , Residues, Residue theorem, Formula for calculating the residue and residue Cauchy Theorem.
14 。 Calculating integrals using residue,Jordan ‘s lemma.
Logarithmic residue, The argument principle, Rouche Theorem, Obtaining maps made by analytic functions, The maximum modulus principle.
8 , Schwarz ‘s lemma and theHadamard three-circle theorem, thePhragmen-Lindeloff theorem, theArzela-Ascoli theorem.
9 , Analytic function spaces,Hurwitz theorem, convergent series of univalent functions. Montel theorem, the meromorphic function space, theRiemann mapping theorem.
10 , Weierstrass factorization theorem, sine functions, factorization,Runge theorem.
18 。 Univalence criterion of local law, inverse of analytic functions, the boundary correspondence principle and basic properties of fractional linear mapping.
19 。 Analytic continuation, analytical functions, complete analytic function of Riemann surfaces and singularities, single-valued theorem.
20 。 A bounded domain of analytic continuation along the symmetry principle and its application of Conformal maps.
11 , Entire functions, order of entire function and type, Weierstrass The product, a meromorphic function, the expansion of meromorphic function in the plane, Connectivity, Mittag-Leffler Theorem, Schwarz Inversion principle.
12 , Function germs, along the roads of analytic continuation, full, single-valued theorem of analytic functions, harmonic functions, maximum principle and minimum principle,Poisson kernel and theHarnark inequality,Harnark theorem.
13 , Raised and subharmonic functions and function,Dirichlet problems andGreen functions.
14 , Jensen formulaPoisson-Jensen formulas,Hadamard factorization theorem.
Complex analysis -2
1 , Bloch theorem,Picard little theorem,Schottky theorem, theMontel-Caratheodory theorem,Picard Conformal mapping theorem, applications in fluid mechanics.
2 , Conformal mapping of the polygonal shape, Pompeiu Formulas, Schwarz-Christoffel Formula.
3 , Gamma function, a meromorphic function Nevanlinna theorems.
Laplace Transformation and gradual progression, asymptotic expansions, Riemann-Zeta Function.
4 , Green formula, double periodicity, elliptic functions andLiouville theorem, factor group,
Elliptic function, Weierstrass Elliptic functions. Jacobi elliptic functionRiemann-Zeta function Riemann-Zeta function Jacobi elliptic functions andJacobi The addition formula for elliptic functions.
5 , Elliptic function field, the elliptic integrals.
6 , Add theorem, elliptic function theory in application to elliptic integrals.
7 , Abel theorem, elliptic modular group.
8 , Mode function,Picard little theorem. Eisentein series. Montel theorem.
9 , Modular Group and basic domains.
10 , Modular forms of algebra,Theta functions of Jacobi transformation formula.
24 。 Riemann existence theorem of uniqueness conditions of Conformal maps.
11 , Congruence group, congruence group die form, continuous form of the integral of a function on a manifold.
12 , Definition of surfaces,Riemann surfaces,Riemann surfaces Rieman metric,Laplace-Beltrami operators, Schwarz-Picktheorem, hyperbolic metric, geodesics.
13 , Hyperbolic isomorphism of discrete groups, basic polygons,Riemann surfaces on the Gauss-Bonnet formula,Riemann-Hurwitz formula.
26 。 Harmonic functions, harmonic functions and analytic functions of contact, infinitely adjustable, and functions, mean value theorem, uniqueness theorem, maximum and minimum principles andLiouville and Harnak theorem,Poisson and Schwarz Integral harmonic functions and series expansions of trigonometric relation.
27 。 Quasiconformal mapping,Dirichlet problem and Conformal mapping in solving the Dirichlet problem in the application. 28。 Harmonic functions and analytic functions applications in fluid mechanics.
Complex Analysis & Riemann Surface
1. Valerian Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable
2. K. Kodaira, Complex Analysis
3. Rudin, Real and complex analysis
4. Gong Sheng, concise complex analysis
1 , A.I.Markushevich And analytic functions on the short tutorial 1987 。
2 , I.I.Privalov , An introduction to complex function, 1984 。
3 , B.V.Shabat , An introduction to complex analysis, volume I, 1986 。
4 , M.A.Lavrentyev 、 B.V.Shabat , Theory of functions of a complex variable method 1987 。
5 , M.A.Evgrafov 、 Yu.B.Sidorov 、 M.V.Fedoryuk 、 M.I.Shabunin 、 K.A.Bezhanov And analytic functions on the problem set,1972 。
6 , T.A.Leontyeva 、 Z.S.Panferov 、 B.C.Serov , Theory of functions of a complex variable problem sets, lecture at the University of Moscow, 1992 。
7 , E.P.Dolzhenko 、 S.N.Nikolaeva , Theory of functions of a complex variable instruction, lectures at the University of Moscow, 1988 。
Supplementary bibliography:
1 , A.V.Bitsadze , Functions of one complex variable basis, 1972 。
2 , A.I.Markushevich , Analytic function theory, 1967 。
3 , V.I.Smirnov Higher mathematics tutorials, 1972 。
4 , A.G.Sveshnikov 、 A.N.Tikhonov , Theory of functions of a complex variable, 1974 。
5 , Y.V.Sidorov 、 M.V.Fedoryuk 、 M.I.Shabunin , Lectures on theory of functions of a complex variable, 1976 。
7 , L.I.Volkovysk 、 G.L.Lunts 、 I.G.Aramanovich , Theory of functions of a complex variable problem sets, 1975 。
Titchmarch ” Function theory”
Ge Luxin ” Theory of functions of a complex variable geometry”
Conway, “Functions of One Complex Variable “
Hormander “An Intro to Complex Analysis in Several Variables”
H.Cartan ” An introduction to analytic function theory ” Preliminary scientific press of the analytic function theory
Beardon, “Complex Analysis”
R.Remmert “Complex Analysis”(GTM,reading in mathematics)
Steven G. Krantz : Function Theory of Several Complex Variables
Steven G. Krantz : Complex Analysis: The Geometric Viewpoint
Lang, Complex analysis :
Elias M. Stein : Complex Analysis
Enterprise Services, the functions of a complex variable course, North
Shi Jihuai, Fundamentals of the theory of functions of several complex variables, higher education
Zhang Nanyue, of the selected topics in theory of functions of a complex variable, North Edition
Ren Yao Fu applied complex analysis, Fudan University Press
Functions of a complex variable in the “classical” analysis “beyond theory, Conformal mapping,
Fan Lili , He Chengqi ” Theory of complex variable functions”
Zhuang ( Yin / Qi ) Thai , He Yuzan ” Theory of complex variable functions ( Feature ?) Selected topics in”
Yu Jiarong of the complex functions of the higher education press
Of the complex functions of the clock spring
J.-P. Serre, “A course of Arithmetics”
O.Forster : Lectures on Riemann Surfaces
Jost : Compact riemann surfaces
Narasimhan : Compact riemann surfaces
Lang : Riemann surfaces ,
Hershel M. Farkas : Riemann Surfaces
143 Department of mathematics of the complex functions of the Dalian group
“The problem sets and books”
145 Examples of higher mathematics and the problem set . Three , Boyarchuk of functions of a complex variable editor
“Increase”
University town also has one of the functions of a complex variable
Serendipities 