本书由在国际上享有盛誉的普林斯顿大学教授Stein等撰写而成, 是一部为数学及相关专业大学二年级和三年级学生编写的教材，理论与实践并重。为了便于非数学专业的学生学习，全书内容简明、易懂，读者只需掌握微积分和线性代数知识。关于本书的详细介绍，请见“影印版前言”。1 q# `, F% X. `5 F; x
本书已被哈佛大学和加利福尼亚理工学院选为教材。与本书相配套的教材《傅立叶分析导论》和《实分析》也已影印出版。
! Q$ m4 ^7 `$ _! g0 g* \ SteIn在国际上享有盛誉，现任美国普林斯顿大学数学系教授。他是当代分析，特别是调和分析领域人物之一。古典调和分析困难问题之一是推广到多维。他是多维欧氏调和分析的创造者之一，为此他发展了许多先进工具，如奇异积分、Radon变换、极大函数等。他还发展了多个实变Foreword
5 ]- h) T, o* v1 G Introductlon
0 B$ k9 H2 W# y& @1 N0 M Chapter 1．Preliminaries to Complex Analysisl Complex numbers and the eompicx plane& X7 \1 a' j* W: o3 e& L/ u
1.1 Basic properties' j \4 b4 {; P2 g- J* Y2 Z6 q
l.2 Convergence
2 J+ o1 c* f3 p& E9 H 1.3 Sets in tim complex plane9 J6 u/ |0 g" D+ p
2 Functions on the complex plane* y% H# l6 k- A) W. K8 e- P
2.l Conltinuous fnetions+ b' L8 @* H7 |% t; F. G3 a
2.2 Holomorphic fimctions7 ^, P7 q+ h3 M" o q1 }& N N' ^! w
2.3 P0weI series
0 ~# {9 ]" k# q6 j$ X8 c 3 Integration along crvcs" |5 P; r$ }) b: s2 Y
4 Exorcises
5 w* V5 ^8 G8 q, K8 V Chapter 2 Cauchy’s Theorem and Its Applications1 Goursat’s theorem
5 v( t% [6 B- n0 D5 {; K7 X 2 Local existencc of primitives and Cauchy s theorem in adisc; B h, M# E' N6 u' T6 |1 a
3 EvaIuatlon of some integrals
% M2 j, w2 H6 l. c( R. a$ q o% I" Z: l 4 Cauchy’s integral formulas
! ^( N; c. y: M8 O1 O- N 5 1lrther applications' u# F, f+ E0 Q! O, y& ^& u5 E4 y
5.1 Morera’s tImorem
% {$ `9 [" L* C. l: \( M# H 5.2 Sequences of holomorphic functions7 |, l5 q2 G# @9 Q C5 w4 ?& R
5.3 Holomorphic functions defined in terms of integrals5.4 Schwarz reflection principle
; j+ d. m, d! M 5.5 Runge’s approxlnlatlon theorem
: p8 j! q3 v& z& a 6 Exereises$ D j7 v* l1 `' K/ S0 y
7 Problems$ U& L1 B8 d$ X
Chapter 3．Meromorphic Functions and the Logarithm1 Zeros and polcs
% t% G# Y: m1 w) \ 2 The residue formuia
/ E; a" w' e8 {' J 2.l Examples
, w j+ ~9 ?+ P# z 3 Singularities and meromorphic functions4 The argmuent principle and applications5 Homotopies and simply connected domains6 The complex logarithm
. ^( W, |9 n1 Q* q2 O9 i3 e 7 Fourier series and harmonic functions6 s9 s t7 m+ ~
8 Exercises
: v5 N. X3 t# Y* r: H/ R 9 Problenis( Y# I0 ^( \5 }2 |; ~& ^/ A
Chapter 4. The Fourier Transforin5 m; p J2 e/ W
1 The class
+ e8 e' g) o$ D5 q L4 M; O3 E( o 2 Action of the Fourier transform on, J* \% l" G' B- E
3 Palev-wiener tbeorem
* t- P0 p8 b5 Q: r, n6 l5 T 4 Exercises
* K* g& v3 j; f8 D2 b 5 Problems1 p; _5 X# N( p! I2 m
Chapter 5. Entire Functions7 D, P6 G. s+ n; d% X3 r. L
1 Jensen's formUla
& I! a* T5 t; Q9 {1 w 2 Functions of finite order* e6 O2 g* Z0 P2 x: G9 |% b
3 Infinite products) U r" y* Y$ \' }4 j ?! H
3.1 Generalities2 g& @( T" w8 I: [
3.2 Example: the product foemula for the sine function4 Weierstrass infinite products4 c0 T: U3 w/ M ]# i
5 Hadamard's factorozatoon theorem
) P; \# H1 }- d7 j 6 Exercises J- x$ k1 s. O) D' {: i9 X6 l. c3 s
7 Problems
: o; }; h" ]6 v+ R1 N; L, v Chapter 6. The Gamma and Zeta Functions, S7 w6 j2 M2 g3 H% I: i) S; [. p
1 The gamma function q' W( }' C. N" Q* F! s
1.1 Analytic continuation
- M+ s' T3 P+ i 1.2 Furtiicr properties of F
6 p) G5 h) C2 G R 2 The zeta function) G! g; R8 E1 ~7 n/ d) a* m4 o8 R% Q
2.1 Functional equation and analytic continuation3 Exercises
0 s; x9 u: P6 q& g8 m 4 Problems
' q( l L. a9 g A Chapter 7. The Zeta Function and Prime Number The-orem1 Zeros of tile zeta function
( f2 O) E. P5 p( w. i$ n hl Esthnates for 1/C(S)
1 j. Q& G, Y& o6 A9 e! p; r 2 Reduction to the functions
1 @' x& Y$ O4 e" k 2.1 Proof of the asymptotics for
& }0 x( x8 J; S* }- w1 k7 y Note on interchanging double sums+ C, `" h# H1 Y' @% ?* U: A# Z
3 Exercises
2 H4 ~* L2 g8 l0 y& j; E7 Z 4 problems- R! \2 n/ j. X( j0 |1 ], x
Chapter 8. Conformal Mappings7 I" ]: h- }6 \& q$ t' O
Chapter 9. An Introduction to Elliptic FunctionsChapter 10. Applications of Theta FunctionsAppendix A: Asymptotics
% Y& S, P; b, g Appendix B: Simple Connectivity and Jordan Curve TheoremNotes and References3 r# ^$ Y# G* z6 c- K' I
Bibliography/ V( r: V0 e+ l: d
Symbol Glossary3 p4 Q2 M: ]6 r2 w. M0 P& \# f
Index
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