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2010 | Buch | 3. Auflage

The Complex Numbers Kapitel lesen Erstes Kapitel lesen

Complex Analysis

verfasst von: Joseph Bak, Donald J. Newman

Verlag: Springer New York

Buchreihe : Undergraduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Beginning with the ?rst edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes inthisedition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtainedby seeing a little more of the “bigpicture”.This includesadditional related results and occasional generalizations that place the results inaslightly broader context. The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the rootsof a generalpolynomial,we added a section on Newton’sMethod,a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative. Aseries of new results relate to the mapping properties of analytic functions. Arevised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the SchwarzRe?ectionPrinciplehasbeenexpandedtoincludere?ectionacrossanalytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations.

Inhaltsverzeichnis

Frontmatter
Chapter 1. The Complex Numbers
Abstract
Numbers of the form \( a + b\sqrt { - 1} \), where a and b are real numbers—what we call complex numbers.appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations. In the 18th century, functions involving complex numberswere found by Euler to yield solutions to differential equations. As more manipulations involving complex numbers were tried, it became apparent that many problems in the theory of real-valued functions could be most easily solved using complex numbers and functions. For all their utility, however, complex numbers enjoyed a poor reputation and were not generally considered legitimate numbers until the middle of the 19th century. Descartes, for example, rejected complex roots of equations and coined the term “gimaginary” for such roots. Euler, too, felt that complex numbers “exist only in the imagination” and considered complex roots of an equation useful only in showing that the equation actually has no solutions.
Joseph Bak, Donald J. Newman
Chapter 2. Functions of the Complex Variable z
Abstract
We wish to examine the notion of a “function of z” where z is a complex variable. To be sure, a complex variable can be viewed as nothing but a pair of real variables so that in one sense a function of z is nothing but a function of two real variables. This was the point of view we took in the last section in discussing continuous functions. But somehow this point of view is too general. There are some functions which are “direct” functions of z = x + iy and not simply functions of the separate pieces x and y.
Joseph Bak, Donald J. Newman
Chapter 3. Analytic Functions
Abstract
The direct functions of z which we have studied so far—polynomials and convergent power series—were shown to be differentiable functions of z. We now take a closer look at the property of differentiability and its relation to the Cauchy-Riemann equations.
Joseph Bak, Donald J. Newman
Chapter 4. Line Integrals and Entire Functions
Abstract
Recall that, according to Theorem 2.9, an everywhere convergent power series represents an entire function.Ourmain goal in the next two chapters is the somewhat surprising converse of that result: namely, that every entire function can be expanded as an everywhere convergent power series. As an immediate corollary, we will be able to prove that every entire function is infinitely differentiable. To arrive at these results, however, we must begin by discussing integrals rather than derivatives.
Joseph Bak, Donald J. Newman
Chapter 5. Properties of Entire Functions
Abstract
We now show that if f is entire and if\( g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ \end{array} } \right. \)then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f. (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.)We begin by showing that the Rectangle Theorem applies to g.
Joseph Bak, Donald J. Newman
Chapter 6. Properties of Analytic Functions
Abstract
In the last two chapters, we studied the connection between everywhere convergent power series and entire functions. We now turn our attention to the more general relationship between power series and analytic functions.According to Theorem 2.9 every power series represents an analytic function inside its circle of convergence. Our first goal is the converse of this theorem: we will show that a function analytic in a disc can be represented there by a power series. We then turn to the question of analytic functions in arbitrary open sets and the local behavior of such functions.
Joseph Bak, Donald J. Newman
Chapter 7. Further Properties of Analytic Functions
Abstract
The Uniqueness Theorem (6.9) states that a non-constant analytic function in a region cannot be constant on any open set. Similarly, according to Proposition 3.7, |f| cannot be constant. Thus a non-constant analytic function cannot map an open set into a point or a circular arc. By applying the Maximum-Modulus Theorem, we can derive the following sharper result on the mapping properties of an analytic function.
Joseph Bak, Donald J. Newman
Chapter 8. Simply Connected Domains
Abstract
As we have seen, it can happen that a function f is analytic on a closed curve C and yet \( \int c f \ne 0 \).
Joseph Bak, Donald J. Newman
Chapter 9. Isolated Singularities of an Analytic Function
Abstract
Introduction While we have concentrated until now on the general properties of analytic functions, we now focus on the special behavior of an analytic function in the neighborhood of an “isolated singularity.”
Joseph Bak, Donald J. Newman
Chapter 10. The Residue Theorem
Abstract
We now seek to generalize the Cauchy Closed Curve Theorem (8.6) to functions which have isolated singularities.
Joseph Bak, Donald J. Newman
Chapter 11. Applications of the Residue Theorem to the Evaluation of Integrals and Sums
Abstract
In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.
Joseph Bak, Donald J. Newman
Chapter 12. Further Contour Integral Techniques
Abstract
We have already seen how the Residue Theorem can be used to evaluate real line integrals. The techniques involved, however, are in noway limited to real integrals. To evaluate an integral along any contour, we can always switch to a more “convenient” contour as long as we account for the pertinent residues of the integrand.
Joseph Bak, Donald J. Newman
Chapter 13. Introduction to Conformal Mapping
Abstract
In this chapter, we take a closer look at themapping properties of an analytic function. Throughout the chapter, all curves z(t) are assumed to be such that \( z(t) \ne 0 \) for all t.
Joseph Bak, Donald J. Newman
Chapter 14. The Riemann Mapping Theorem
Abstract
Before proving the Riemann Mapping Theorem, we examine the relation between conformal mapping and the theory of fluid flow. Our main goal is to motivate some of the results of the next section and the treatment here will be less formal than that of the remainder of the book.
Joseph Bak, Donald J. Newman
Chapter 15. Maximum-Modulus Theorems for Unbounded Domains
Abstract
The Maximum-Modulus Theorem (6.13) shows that a function which is C-analytic in a compact domain D assumes its maximum modulus on the boundary. In general, if we consider unbounded domains, the theorem no longer holds. For example, \( f(z) = e^z \) is analytic and unbounded in the right half-plane despite the fact that on the boundary \( |e^z | = e^{iy} | = 1 \). Nevertheless, given certain restrictions on the growth of the function, we can conclude that it attains its maximum modulus on the boundary. The most natural such condition is that the function remain bounded throughout D.
Joseph Bak, Donald J. Newman
Chapter 16. Harmonic Functions
Abstract
In this chapter, we focus on the real parts of analytic functions and their connection with real harmonic functions.
Joseph Bak, Donald J. Newman
Chapter 17. Different Forms of Analytic Functions
Abstract
The analytic functions we have encountered so far have generally been defined either by power series or as a combination of the elementary polynomial, trigonometric and exponential functions, alongwith their inverse functions. In this chapter, we consider three different ways of representing analytic functions. We begin with infinite products and then take a closer look at functions defined by definite integrals, a topic touched upon earlier in Chapter 7 and in Chapter 12.2. Finally, we define Dirichlet series, which provide a link between analytic functions and number theory.
Joseph Bak, Donald J. Newman
Chapter 18. Analytic Continuation; The Gamma and Zeta Functions
Abstract
Suppose we are given a function f which is analytic in a region D. We will say that f can be continued analytically to a region D 1 that intersects D if there exists a function g, analytic in D 1 and such that g = f throughout \( D_1 \cap D_2 \). By the Uniqueness Theorem (6.9) any such continuation of f is uniquely determined. (It is possible, however, to have two analytic continuations g 1 and g 2 of a function f to regions D 1 and D 2 respectively with \( g_1 \ne g_2 \) throughout \( D_1 \cap D_2 \). See Exercise 1.)
Joseph Bak, Donald J. Newman
Chapter 19. Applications to Other Areas of Mathematics
Abstract
We have already seen, especially in Chapter 11, howthe methods of complex analysis can be applied to the solution of problems from other area of mathematics. In this chapter we will get some insight into the fantastic breadth of such applications. For that reason, the topics chosen are rather disparate. Section 19.1 involves calculating the total variation of a real function, and illustraties how the methods of Chapter 11 can be applied to yet another nontypical problem. Section 19.2 offers a proof of the classic Fourier Uniqueness Theorem using two preliminary results from real analysis and a surprising application of Liouville's theorem. In Section 19.3 we see how the use of a generating function allows complex analytic results to be applied to an infinite system of (real) equations. Generating functions are also the key to the four different problems in number theory that comprise section 19.4. Finally, in section 19.5, we offer a well-trimmed analytic proof of the prime number theorem based on properties of the Zeta function and another Dirichlet series.
Joseph Bak, Donald J. Newman
Backmatter
Metadaten
Titel
Complex Analysis
verfasst von
Joseph Bak
Donald J. Newman
Copyright-Jahr
2010
Verlag
Springer New York
Electronic ISBN
978-1-4419-7288-0
Print ISBN
978-1-4419-7287-3
DOI
https://doi.org/10.1007/978-1-4419-7288-0

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