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E-raamat: Introduction to Complex Analysis

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Introduction to Complex AnalysisSuurem pilt
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In this text, the reader will learn that all the basic functions that arise in calculus--such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet--are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here.

This book could serve for either a one-semester course or a two-semester course in complex analysis for beginning graduate students or for well-prepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus.
Preface vii
Some basic notation xiii
Chapter 1 Basic calculus in the complex domain
1(60)
§1.1 Complex numbers, power series, and exponentials
3(9)
Exercises
12(2)
§1.2 Holomorphic functions, derivatives, and path integrals
14(10)
Exercises
24(2)
§1.3 Holomorphic functions defined by power series
26(6)
Exercises
32(2)
§1.4 Exponential and trigonometric functions: Euler's formula
34(7)
Exercises
41(3)
§1.5 Square roots, logs, and other inverse functions
44(4)
Exercises
48(10)
§1.6 Pi is irrational
58(3)
Chapter 2 Going deeper - the Cauchy integral theorem and consequences
61(74)
§2.1 The Cauchy integral theorem and the Cauchy integral formula
63(8)
Exercises
71(2)
§2.2 The maximum principle, Liouville's theorem, and the fundamental theorem of algebra
73(3)
Exercises
76(2)
§2.3 Harmonic functions on planar domains
78(9)
Exercises
87(2)
§2.4 Morera's theorem, the Schwarz reflection principle, and Goursat's theorem
89(3)
Exercises
92(1)
§2.5 Infinite products
93(13)
Exercises
106(1)
§2.6 Uniqueness and analytic continuation
107(5)
Exercises
112(2)
§2.7 Singularities
114(3)
Exercises
117(1)
§2.8 Laurent series
118(4)
Exercises
122(1)
§2.9 Green's theorem
123(5)
§2.10 The fundamental theorem of algebra (elementary proof)
128(2)
§2.11 Absolutely convergent series
130(5)
Chapter 3 Fourier analysis and complex function theory
135(48)
§3.1 Fourier series and the Poisson integral
137(13)
Exercises
150(2)
§3.2 Fourier transforms
152(7)
Exercises
159(1)
More general sufficient condition for / E A(M)
160(2)
Fourier uniqueness
162(1)
§3.3 Laplace transforms and Mellin transforms
163(2)
Exercises
165(2)
The matrix Laplace transform and Duhamel's formula
167(2)
§3.4 Inner product spaces
169(3)
§3.5 The matrix exponential
172(2)
§3.6 The Weierstrass and Runge approximation theorems
174(9)
Chapter 4 Residue calculus, the argument principle, and two very special functions
183(60)
§4.1 Residue calculus
186(7)
Exercises
193(3)
§4.2 The argument principle
196(5)
Exercises
201(2)
§4.3 The Gamma function
203(4)
Exercises
207(2)
The Legendre duplication formula
209(2)
§4.4 The Riemann zeta function and the prime number theorem
211(8)
Counting primes
219(2)
The prime number theorem
221(4)
Exercises
225(2)
§4.5 Euler's constant
227(6)
§4.6 Hadamard's factorization theorem
233(10)
Chapter 5 Conformal maps and geometrical aspects of complex function theory
243(80)
§5.1 Conformal maps
246(8)
Exercises
254(1)
§5.2 Normal families
255(1)
Exercises
256(1)
§5.3 The Riemann sphere and other Riemann surfaces
257(8)
Exercises
265(3)
§5.4 The Riemann mapping theorem
268(3)
Exercises
271(1)
§5.5 Boundary behavior of conformal maps
272(3)
Exercises
275(2)
§5.6 Covering maps
277(2)
Exercises
279(1)
§5.7 The disk covers the twice-punctured plane
280(1)
Exercises
281(3)
§5.8 Montel's theorem
284(2)
Exercises on Fatou sets and Julia sets
286(3)
§5.9 Picard's theorem
289(1)
Exercises
290(1)
§5.10 Harmonic functions II
290(12)
Exercises
302(1)
§5.11 Surfaces and metric tensors
303(8)
§5.12 Poincare metrics
311(7)
§5.13 Groups
318(5)
Chapter 6 Elliptic functions and elliptic integrals
323(40)
§6.1 Periodic and doubly periodic functions
325(3)
Exercises
328(3)
§6.2 The Weierstrass P-function in elliptic function theory
331(3)
Exercises
334(3)
§6.3 Theta functions and elliptic functions
337(4)
Exercises
341(1)
§6.4 Elliptic integrals
342(6)
Exercises
348(2)
§6.5 The Riemann surface of the square root of a cubic
350(5)
Exercises
355(3)
§6.6 Rapid evaluation of the Weierstrass P-function
358(1)
Rectangular lattices
359(4)
Chapter 7 Complex analysis and differential equations
363(54)
§7.1 Bessel functions
366(15)
Exercises
381(1)
§7.2 Differential equations on a complex domain
382(21)
Exercises
403(2)
§7.3 Holomorphic families of differential equations
405(5)
Exercises
410(2)
§7.4 From wave equations to Bessel and Legendre equations
412(5)
Appendix A Complementary material
417(56)
§A.1 Metric spaces, convergence, and compactness
418(12)
Exercises
430(1)
§A.2 Derivatives and diffeomorphisms
431(6)
§A.3 The Laplace asymptotic method and Stirling's formula
437(4)
§A.4 The Stieltjes integral
441(7)
§A.5 Abelian theorems and Tauberian theorems
448(11)
§A.6 Cubics, quartics, and quintics
459(14)
Bibliography 473(4)
Index 477
Michael E. Taylor, University of North Carolina, Chapel Hill, NC.
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