Complex Analysis

Complex Analysis

2.0 2
by Elias M. Stein, Rami Shakarchi

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With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of

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With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle.

With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory.

Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences.

The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

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Complex Analysis

By Elias M. Stein Rami Shakarchi

Princeton University Press

Copyright © 2003 Princeton University Press
All right reserved.

ISBN: 0-691-11385-8

Chapter One

Preliminaries to Complex Analysis

The sweeping development of mathematics during the last two centuries is due in large part to the introduction of complex numbers; paradoxically, this is based on the seemingly absurd notion that there are numbers whose squares are negative. E. Borel, 1952

This chapter is devoted to the exposition of basic preliminary material which we use extensively throughout of this book.

We begin with a quick review of the algebraic and analytic properties of complex numbers followed by some topological notions of sets in the complex plane. (See also the exercises at the end of Chapter 1 in Book I.)

Then, we define precisely the key notion of holomorphicity, which is the complex analytic version of differentiability. This allows us to discuss the Cauchy-Riemann equations, and power series.

Finally, we define the notion of a curve and the integral of a function along it. In particular, we shall prove an important result, which we state loosely as follows: if a function f has a primitive, in the sense that there exists a function F that is holomorphic and whose derivative is precisely f, then for any closed curve [gamma]

[[integral].sub.[gamma]] f(z) dz = 0.

This is the first step towards Cauchy's theorem, which plays a central role in complex function theory.

1 Complex numbers and the complex plane

Many of the facts covered in this section were already used in Book I.

1.1 Basic properties

A complex number takes the form z = x + iy where x and y are real, and i is an imaginary number that satisfies [i.sup.2] = -1. We call x and y the real part and the imaginary part of z, respectively, and we write

x = Re(z) and y = Im(z).

The real numbers are precisely those complex numbers with zero imaginary parts. A complex number with zero real part is said to be purely imaginary.

Throughout our presentation, the set of all complex numbers is denoted by C. The complex numbers can be visualized as the usual Euclidean plane by the following simple identification: the complex number z = x + iy [member of] C is identified with the point (x; y) [member of] [R.sup.2]. For example, 0 corresponds to the origin and i corresponds to (0, 1). Naturally, the x and y axis of [R.sup.2] are called the real axis and imaginary axis, because they correspond to the real and purely imaginary numbers, respectively. (See Figure 1.)

The natural rules for adding and multiplying complex numbers can be obtained simply by treating all numbers as if they were real, and keeping in mind that [i.sup.2] = -1. If [z.sub.1] = [x.sub.1] + i[y.sub.1] and [z.sub.2] = [x.sub.2] + i[y.sub.2], then

[z.sub.1] + [z.sub.2] = ([x.sub.1] + [x.sub.2]) + i([y.sub.1] + [y.sub.2]); and also

[z.sub.1][z.sub.2] = ([x.sub.1] + i[y.sub.1])([x.sub.2] + i[y.sub.2]) = [x.sub.1][x.sub.2] + i[x.sub.1][y.sub.2] + i[y.sub.1][x.sub.2] + [i.sup.2][y.sub.1][y.sub.2] = ([x.sub.1][x.sub.2] - [y.sub.1][y.sub.2]) + i([x.sub.1][y.sub.2] + [y.sub.1][x.sub.2]).

If we take the two expressions above as the definitions of addition and multiplication, it is a simple matter to verify the following desirable properties:

Commutativity: [z.sub.1] + [z.sub.2] = [z.sub.2] + [z.sub.1] and [z.sub.1][z.sub.2] = [z.sub.2][z.sub.1] for all [z.sub.1], [z.sub.2] [member of] C.

Associativity: ([z.sub.1] + [z.sub.2]) + [z.sub.3] = [z.sub.1] + ([z.sub.2] + [z.sub.3]); and ([z.sub.1][z.sub.2])[z.sub.3] = [z.sub.1]([z.sub.2][z.sub.3]) for [z.sub.1], [z.sub.2], [z.sub.3], [member of] C.

Distributivity: [z.sub.1]([z.sub.2] + [z.sub.3]) = [z.sub.1][z.sub.2] + [z.sub.1][z.sub.3] for all [z.sub.1], [z.sub.2], [z.sub.3] [member of] C.

Of course, addition of complex numbers corresponds to addition of the corresponding vectors in the plane [R.sup.2]. Multiplication, however, consists of a rotation composed with a dilation, a fact that will become transparent once we have introduced the polar form of a complex number. At present we observe that multiplication by i corresponds to a rotation by an angle of [pi]/2.

The notion of length, or absolute value of a complex number is identical to the notion of Euclidean length in [R.sup.2]. More precisely, we define the absolute value of a complex number z = x + iy by

|z| = [([x.sup.2] + [y.sup.2])].sup.1/2],

so that [absolute value of z] is precisely the distance from the origin to the point (x, y). In particular, the triangle inequality holds:

|z + w| [less than or equal to] |z| + |w| for all z, w [member of] C.

We record here other useful inequalities. For all z [member of] C we have both |Re(z)| [less than or equal to] |z| and |Im(z)| [less than or equal to] |z|, and for all z, w [member of] C

||z| - ||w| [less than or equal to] |z - w|.

This follows from the triangle inequality since

|z| [less than or equal to] |z - w| + |w| and |w| [less than or equal to] |z - w| + |z|.

The complex conjugate of z = x + iy is defined by

[bar.z] = x - iy,

and it is obtained by a reflection across the real axis in the plane. In fact a complex number z is real if and only if z = [bar.z], and it is purely imaginary if and only if z = -[bar.z].

The reader should have no difficulty checking that

Re(z) = z + [bar.z]/2 and Im(z) = z - [bar.z]/2i.

Also, one has

||z|.sup.2] = z[bar.z] and as a consequence 1/z = [bar.z] / ||z|.sup.2] whenever z [not equal to] 0.

Any non-zero complex number z can be written in polar form

z = [re.sup.i[theta]],

where r > 0; also [theta] [member of] R is called the argument of z (defined uniquely up to a multiple of 2[pi]) and is often denoted by arg z, and

[e.sup.i[theta]] = cos [theta] + i sin [theta].

Since |[e.sup.i[theta]| = 1 we observe that r = |z|, and [theta] is simply the angle (with positive counterclockwise orientation) between the positive real axis and the half-line starting at the origin and passing through z. (See Figure 2.)

Finally, note that if z = r[e.sup.i[theta]] and w = s[e.sup.i[??]], then

zw = rs[e.sup.i([theta]+[??])],

so multiplication by a complex number corresponds to a homothety in [R.sup.2] (that is, a rotation composed with a dilation).

1.2 Convergence

We make a transition from the arithmetic and geometric properties of complex numbers described above to the key notions of convergence and limits.

A sequence {[z.sub.1], [z.sub.2], ...} of complex numbers is said to converge to w [member of] C if


This notion of convergence is not new. Indeed, since absolute values in C and Euclidean distances in [R.sup.2] coincide, we see that [z.sub.n] converges to w if and only if the corresponding sequence of points in the complex plane converges to the point that corresponds to w.

As an exercise, the reader can check that the sequence {[z.sub.n]} converges to w if and only if the sequence of real and imaginary parts of [z.sub.n] converge to the real and imaginary parts of w, respectively.

Since it is sometimes not possible to readily identify the limit of a sequence (for example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), it is convenient to have a condition on the sequence itself which is equivalent to its convergence. A sequence {[z.sub.n]} is said to be a Cauchy sequence (or simply Cauchy) if

|[z.sub.n] - [z.sub.m]| [right arrow] 0 as n, m [right arrow] [infinity].

In other words, given [??] > 0 there exists an integer N > 0 so that [absolute value of [z.sub.n] - [z.sub.m]] < [??] whenever n, m > N. An important fact of real analysis is that R is complete: every Cauchy sequence of real numbers converges to a real number. Since the sequence {[z.sub.n]} is Cauchy if and only if the sequences of real and imaginary parts of [z.sub.n] are, we conclude that every Cauchy sequence in C has a limit in C. We have thus the following result.

Theorem 1.1 C, the complex numbers, is complete.

We now turn our attention to some simple topological considerations that are necessary in our study of functions. Here again, the reader will note that no new notions are introduced, but rather previous notions are now presented in terms of a new vocabulary.

1.3 Sets in the complex plane

If [z.sub.0] [member of] C and r > 0, we define the open disc [D.sub.r]([z.sub.0]) of radius r centered at [z.sub.0] to be the set of all complex numbers that are at absolute value strictly less than r from [z.sub.0]. In other words,

[D.sub.r]([z.sub.0]) = {z [member of] C : |z - [z.sub.o]| < r},

and this is precisely the usual disc in the plane of radius r centered at [z.sub.0]. The closed disc [bar.[D.sub.r]]([z.sub.0]) of radius r centered at [z.sub.0] is defined by


and the boundary of either the open or closed disc is the circle

[C.sub.r]([z.sub.0]) = {z [member of] C : |z - [z.sub.0]| = r}.

Since the unit disc (that is, the open disc centered at the origin and of radius 1) plays an important role in later chapters, we will often denote it by D,

D = {z [member of] C : |z| < 1}.

Given a set [OMEGA] [subset] C, a point [z.sub.0] is an interior point of [OMEGA] if there exists r > 0 such that

[D.sub.r]([z.sub.0]) [subset] [OMEGA].

The interior of [OMEGA] consists of all its interior points. Finally, a set [OMEGA] is open if every point in that set is an interior point of [OMEGA]. This definition coincides precisely with the definition of an open set in [R.sub.2].

A set [OMEGA] is closed if its complement [[OMEGA].sup.c] = C - [OMEGA] is open. This property can be reformulated in terms of limit points. A point z [member of] C is said to be a limit point of the set [OMEGA] if there exists a sequence of points [z.sub.n] [member of] [OMEGA] such that [z.sub.n] [not equal to] z and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The reader can now check that a set is closed if and only if it contains all its limit points. The closure of any set [OMEGA] is the union of [OMEGA] and its limit points, and is often denoted by [bar.[OMEGA]].

Finally, the boundary of a set [OMEGA] is equal to its closure minus its interior, and is often denoted by [partial derivative][OMEGA].

A set [OMEGA] is bounded if there exists M > 0 such that |z| < M whenever z [member of] [OMEGA]. In other words, the set [OMEGA] is contained in some large disc. If [OMEGA] is bounded, we define its diameter by


A set [OMEGA] is said to be compact if it is closed and bounded. Arguing as in the case of real variables, one can prove the following.

Theorem 1.2 The set [OMEGA] [subset] C is compact if and only if every sequence {[z.sub.n]} [subset] [OMEGA] has a subsequence that converges to a point in [OMEGA].

An open covering of [OMEGA] is a family of open sets {[U.sub.[alpha]]} (not necessarily countable) such that


In analogy with the situation in R, we have the following equivalent formulation of compactness.

Theorem 1.3 A set [OMEGA] is compact if and only if every open covering of [OMEGA] has a finite subcovering.

Another interesting property of compactness is that of nested sets. We shall in fact use this result at the very beginning of our study of complex function theory, more precisely in the proof of Goursat's theorem in Chapter 2.

Proposition 1.4 If [[OMEGA].sub.1] [contains] [[OMEGA].sub.2] [contains] ... [contains] [[OMEGA].sub.n] [contains] ... is a sequence of non-empty compact sets in C with the property that

diam([[OMEGA].sub.n]) [right arrow] 0 as n [right arrow] [infinity],

then there exists a unique point w [member of] C such that w [member of] [[OMEGA].sub.n] for all n.

Proof. Choose a point [z.sub.n] in each [[OMEGA].sub.n]. The condition diam([[OMEGA].sub.n]) [right arrow] 0 says precisely that {[z.sub.n]} is a Cauchy sequence, therefore this sequence converges to a limit that we call w. Since each set [[OMEGA].sub.n] is compact we must have w [member of] [[OMEGA].sub.n] for all n. Finally, w is the unique point satisfying this property, for otherwise, if w' satisfied the same property with w' [not equal to] w we would have |w - w'| > 0 and the condition diam([[OMEGA].sub.n]) [right arrow] 0 would be violated.

The last notion we need is that of connectedness. An open set [OMEGA] [subset] C is said to be connected if it is not possible to find two disjoint non-empty open sets [[OMEGA].sub.1] and [[OMEGA].sub.2] such that

[OMEGA] = [[OMEGA].sub.1] [union] [[OMEGA].sub.2].

A connected open set in C will be called a region. Similarly, a closed set F is connected if one cannot write F = [F.sub.1] [union] [F.sub.2] where [F.sub.1] and [F.sub.2] are disjoint non-empty closed sets.

There is an equivalent definition of connectedness for open sets in terms of curves, which is often useful in practice: an open set [OMEGA] is connected if and only if any two points in [OMEGA] can be joined by a curve [gamma] entirely contained in [OMEGA]. See Exercise 5 for more details.

2 Functions on the complex plane

2.1 Continuous functions

Let f be a function defined on a set [OMEGA] of complex numbers. We say that f is continuous at the point [z.sub.0] [member of] [OMEGA] if for every [??] > 0 there exists [delta] > 0 such that whenever z [member of] [OMEGA] and |z - [z.sub.0]| < [delta] then |f(z) - f([z.sub.0])| < [??]. An equivalent definition is that for every sequence {[z.sub.1], [z.sub.2], ...} [subset] [OMEGA] such that lim [z.sub.n] = [z.sub.0], then lim f([z.sub.n]) = f([z.sub.0]).

The function f is said to be continuous on [OMEGA] if it is continuous at every point of [OMEGA]. Sums and products of continuous functions are also continuous.

Since the notions of convergence for complex numbers and points in [R.sub.2


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