Complex Variables

Preface

This book originated at the University of Pennsylvania in the Spring of 1968 as a set of lecture notes addressed to undergraduate math and science majors. It is intended for an introductory one-semester or quarter-and-a-half course with minimal prerequisites; it is neither a reference nor a handbook.

We approach complex analysis via real plane calculus, Green’s Theorem and the Green’s identities, determination by boundary values, harmonic functions, and steady-state temperatures. The conscientious student will compute many line integrals and directional derivatives as he works through the early chapters. The beautiful Cauchy theory for complex analytic functions is preceded by its harmonic counterpart.

The young student is likely to assume that an arbitrary differentiable function defined somewhere enjoys the remarkable properties of complex analytic functions. From the beginning we stress that

  (i) the analytic f(z) = u(x, y) + iυ(x, y) is much better behaved than the arbitrary function encountered in freshman calculus or the first , course;

 (ii)   this is because u(x, y), υ(x, y) satisfy certain basic partial differential equations;

(iii)   one can obtain much useful information about solutions of such equations without actually solving them.

In developing integration theory, we emphasize the analytic aspects at the expense of the topological or combinatorial. Thus, a complex function f(z) is defined to be analytic at a point if it is continuously complex differentiable in a neighborhood of that point. The Cauchy Integral Theorem is thereby an easy consequence of Green’s Theorem and the Cauchy-Riemann equations. Goursat’s remarkable deepening of the Integral Theorem is discussed, but is not proved. On the other hand, we make much of the standard techniques of representing a function as an integral and then bounding that integral (the ML-inequality) or differentiating under the integral sign. The integral representation formulas (Green’s Third Identity, the Poisson Integral, the Cauchy Formula) are the true heroes of these chapters.

The second half of the book (Chapters 5–8) is motivated by two concerns: the integration of functions which possess singularities, and the behavior of analytic mappings w = f(z). Power series are developed first; thence flows the basic factorization from this comes all the rest. The book concludes with a discussion (no proof) of the Riemann Mapping Theorem.

The author recalls with pleasure many, many hours spent discussing complex analysis with Professor Jerry Kazdan at the University of Pennsylvania and nearby spots. Particular thanks are due Professor Kazdan and Professor Bob Hall for reading the manuscript and making many usable suggestions. Finally, the author is happy to record his gratitude to the staff of Allyn and Bacon for encouragement and prompt technical assistance over the months and miles.

FRANCIS J. FLANIGAN

COMPLEX VARIABLES

Harmonic and Analytic Functions

1

Calculus in the Plane

Section 1.1 DOMAINS IN THE xy-PLANE

1.1.0 Introduction

Here’s what we’ll do in the first few chapters:

1. We examine the geography of the xy-plane. Some of this will be familiar from basic calculus (for example, distance between points), some may be new to you (for example, the important notion of domain). We must also consider curves in the plane.

2. We consider real-valued functions u(x, y) defined in the plane. We will examine the derivatives (partial derivatives, gradient, directional derivatives) and integrals (line integrals, double integrals) of these functions. Most of (1) above will be necessary for (2). All this happens in this chapter.

3. We next focus attention on a particular kind of real-valued function u(x, y), the so-called harmonic function (Chapter 2). These are very interesting in their own right, have beautiful physical interpretations, and point the way to complex analytic functions.

4. At last (Chapter 3) we consider points (x, y) of the plane as complex numbers x + iy and we begin our study of complex-valued functions of a complex variable. This study occupies the rest of the book.

One disadvantage of this approach is the fact that complex numbers and complex analytic functions (our chief topic) do not appear until the third chapter. Admittedly, it would be possible to move directly from step (1) to step (4), making only brief reference to real-valued functions. On the other hand, the present route affords us

 (i) a good look at some very worthwhile two-variable real calculus, and

(ii) an insight into the reasons behind some of the magical properties of complex analytic functions, which (as we will see) flow from (a) the natural properties of real-valued harmonic functions u(x, y) and (b) the fact that we can multiply and divide points in the plane. In the present approach the influences (a) and (b) will be considered separately before being combined.

One effect we hope for: You will learn to appreciate the difference between a complex analytic function (roughly, a complex-valued function f(z) having a complex derivative f′(z)) and the real functions y = f(x) which you differentiated in calculus. Don’t be deceived by the similarity of the notations f(z), f(x). The complex analytic function f(z) turns out to be much more special, enjoying many beautiful properties not shared by the run-of-the-mill function from ordinary real calculus. The reason (see (a) above) is that f(x) is merely f(x), whereas the complex analytic function f(z) can be written as

where z = x + iy and u(x, y), υ(x, y) are each real-valued harmonic functions related to each other in a very strong way: the Cauchy–Riemann equations

In summary, the deceptively simple hypothesis that

forces a great deal of structure on f(z); moreover, this structure mirrors the structure of the harmonic u(x, y) and υ(x, y), functions of two real variables.

All these comments will make more sense after you have read Chapter 4. Let us begin now at the beginning.

1.1.1 The Algebraic Structure in ²

Throughout these pages denotes the set of all real numbers. By ² (read -two) we mean the set of all ordered pairs (x, y) with both x and y in . Ordered pair means (x, y) = (x1, y1) if and only if x = x1 and y = y1. We call these pairs (x, y) points. Some points in ² are (3, −2), (1, 0), (0, 1), ( , −1), (0, 0). It is customary to denote the typical point (x, y) by z; thus, z = (x, y). We’ll also use z0 = (x0, y0) and = ( , ). Here, , , are the lower-case Greek letters zeta, xi, eta, respectively.

We may add and subtract points in ². Thus, if z = (x, y) and = ( , ), we define

For example, if z = (1, 2) and = (3, −1), then

We may also multiply a point of ² by a number in . Thus, if z = (x, y) and c , we define

For example, if z = (1, 2) and c = 5, then

Note how strongly the definitions of addition, subtraction, and multiplication in ² depend on the corresponding operations in the real numbers themselves.

Preview

In Chapter 3, we will define the product z of points z and in ² (note that in the product cz above, the factor c was required to be in ). When this new product is defined, the set ² will be called the complex numbers and thereafter denoted by . Although it would be easy to define the product z now, we feel it is more instructive and dramatic to squeeze as much as we can from the familiar real calculus first.

Pictures

The representation of ² as the xy-plane should be familiar to you. It is standard to denote the origin (0, 0) by 0. This leads to no confusion, as we will see. Note that if we draw line segments from 0 to z and 0 to , then the sum z + is the fourth corner of the parallelogram determined by the two segments. You should examine Figure 1.1 (geometric!) and the definition of z + (algebraic!) until convinced of this.

Figure 1.1

Exercises to Paragraph 1.1.1

1. Let z = (2, −2), = (−1, 5). Compute

(a) z + ,

(b) − z,

(c) 2z − ,

(d) z + 4 .

2. Given z, as in Exercise 1, solve the following for w = (u, υ):

(a) z + 2 + 3w = 0,

(b) 2z + w = − .

1.1.2 The Distance Structure in ²

Let z = (x, y) ². We define the norm (or length, modulus, absolute value) of z (denoted |z|) by

For example, if z = (1, −2), then .

Note that the norm is a nonnegative real number (square root!), |z| ≥ 0, and, in fact, |z| = 0 if and only if z = 0 (= (0, 0)). The definition of |z| agrees with the famous Pythagorean theorem for right triangles,

as Figure 1.2 shows.

Now we use the above to define the distance between z = (x, y) and z0 = (x0, y0) as the norm of their difference z − z0; that is, distance from z to z0 = |z − z0|. See Figure 1.3. Since z − z0 = (x − x0, y − y0), we have the formula

Figure 1.2

Figure 1.3

For example, if z = (1, −2) and z0 = (2, 5), then the distance from z to z0 is

Exercises to Paragraph 1.1.2

1. Let z = (−1, 4), z0 = (2, 2). Compute:

(a) |z|,

(b) |z0|,

(c) |z − z0|,

(d) |z0 − z|.

2. Compute the distance from z0 to z, with z, z0 as in Exercise 1.

3. Sketch in the plane the sets of points z determined by each of the following conditions. Here, z0 = (1, 1).

(a) |z| = 1.

(b) |z| < 1.

(c) |z − z0| = 1.

(d) |z − z0| ≥ 1.

4. Establish the following useful inequalities. Sketch!

(a) |z + | ≤ |z| + | |        (triangle inequality).

(b) |x| ≤ |z|, |y| ≤ |z|        where z = (x, y).

1.1.3 Domains in ²

We intend to define a domain to be an open connected subset of ². Hence, we must first make sense of the words open and connected. The notion of distance will be crucial for this.

Let z0 be a fixed point in ² and r > 0 a given positive number. We denote by D(z0; r) the disc of radius r centered at z0, defined as the set

Read this as follows: D(z0; r) equals the set of all points z in ² such that the distance |z − z0| is less than r. Note that its rim, the circle of points whose distance from z0 is exactly r, is not included in D(z0; r).

Now let Ω be any subset of ², and suppose z Ω (z is in Ω). See Figure 1.4. We say that z is an interior point of Ω if and only if there exists a disc centered at z and contained entirely inside Ω; that is, if and only if there exists r > 0 such that D(z; r) Ω.

Figure 1.4

Examples of Interior Points

1. Let Ω = ². Then every point z is an interior point of ² because any disc around z will surely be contained in ².

2. Let Ω be itself a disc, Ω = D(z0; r). If z is any point of D(z0; r), then we may find a smaller disc D(z; r1) contained inside D(z0; r); see Figure 1.5. Thus, every point of D(z0; r) is an interior point of D(z0; r).

3. This time let Ω consist of a disc D(z0; r) together with its rim, the set of points z satisfying |z − z0| = r. Now not all points of Ω are interior points. More precisely, the points of D(z0; r) are interior points of Ω (why?), but no point z on the rim of Ω is an interior point because we cannot surround such a point (on the borderline) with a disc that fits inside Ω. Perfectly reasonable. See Figure 1.6.

Figure 1.5

Figure 1.6

At last we may define open set. A subset Ω of ² is open if and only if every point of Ω is an interior point of Ω. Thus, every point of an open set is well inside the set; none of its points are on its boundary.

Examples of Open Sets

1. The entire plane ² is open.

2. The empty set , the set with no points, is open. Since there are no points, we needn’t worry about discs around them.

3. Each disc D(z0; r) is open. From now on we may refer to these as open discs, to distinguish them from discs with rims included (closed discs).

4. The upper half-plane, the set of all z = (x, y) satisfying y > 0, is an open set. It is essential for openness here that we do not include any points on the x-axis (y = 0).

5. Let Ω be the set D(z0; r) − {z0}, that is, the disc with the center point z0 removed (called the punctured disc). You should convince yourself that this set Ω is open. More generally, if we take any open set S and remove a point z, the new set S − {z} is again open.

You should be able to give an example of a set that is not open (see the third example of interior points under the preceding topic (Figure 1.6)).

In case you are wondering, a subset S of ² is closed if and only if its complement ² − S (the set of points in ² but not in S) is open. One example of a closed set is ², since its complement ² − ² is the empty set , which is open. Similarly, the empty set is closed (why?). It can be shown that the only subsets of ² which are both open and closed are ² and the empty set . Some sets are neither open nor closed.

Another important example of a closed set is the following: Let (z0; r) denote the set of points whose distance from z0 is less than or equal to the positive number r; that is,

This is the closed disc (rim included) of radius r centered at z0. We discussed this set in Example 3 under interior points. See Figure 1.6. You should convince yourself that it is closed in the sense that its complement ² − (z0; r) is open.

Now we continue our definition of domain. We must define what we mean for an open set to be connected. Actually, it is possible to define connectedness for any subset of ², not just the open subsets, but this takes more work and would be superfluous at the moment.

Let Ω be an open subset of ²; Ω is disconnected if and only if there exist nonempty open sets Ω1, Ω2 which are disjoint (no points in common, Ω1 Ω2 = ) and whose union is Ω. Thus, a disconnected open set Ω may be decomposed into two smaller, nonoverlapping, nonempty, open sets. An open set Ω is connected if and only if it is not disconnected. See Figure 1.7.

Challenge

You might try to prove that an open set Ω is connected if and only if any two points in Ω may be linked by a path made of a finite number of straight-line segments lying entirely in Ω.

Thus, there are two equivalent notions of connectedness for an open set Ω: (1) it can’t be broken into disjoint open pieces, or (2) any pair of points may be linked by a path in Ω. Both are reasonable.

Figure 1.7

At last we make our basic definition. A subset Ω of ² is a domain if and only if it is open and connected. The domains we will encounter most frequently are ² itself, the open discs D(z0; r), the punctured discs, and the upper half-plane (see the fourth example under the preceding topic). Not all domains, of course, are quite so symmetric as these; this is fortunate or unfortunate, depending on your point of view.

Exercises to Paragraph 1.1.3

1. (a) Sketch the set S of points z = (x, y) satisfying x ≥ 0.

(b) Verify that the subset of interior points of S is determined by the condition x > 0.

(c) Is the subset in (b) a domain?

2. (a) Sketch the annulus Ω = {z | 1 < |z| < 2}.

(b) There is a hole in Ω. Is Ω connected?

(c) Verify that Ω is a domain.

3. Rather than designate one of the standard domains by a capital letter, we often speak of "the unit disc |z| < 1, the punctured disc 0 < |z| < 1," and so on. Sketch the sets determined by each of the following conditions and decide which are domains. Here, z0 is an arbitrary but fixed point.

(a) |z| > 1.

(b) 1 ≤ |z| ≤ 2.

(c) |z − z0| < 1.

(d) |z − z0| ≤ 2.

4. Let Ω be a domain and let S be a nonempty subset of Ω satisfying (i) S is open, (ii) its complement Ω − S is open (sometimes stated as S is closed in Ω). Prove S = Ω; that is, Ω − S is empty.

5. Challenge question. Let Ω be open. Prove that Ω is connected if and only if any two points in Ω may be linked by a path consisting of a finite number of straight-line segments lying entirely in Ω.

Hint: Given Ω connected and z0 Ω, prove that the subset S consisting of all points of Ω which may be linked to z0 by the specified type of path is not empty (clear!), open, and also closed in Ω. By Exercise 4, S = Ω, whence any two points of Ω may be linked. Constructing the set S (the points for which what you wish to prove is true) is an important method in dealing with connectedness.

6. Is a disc a circle?

1.1.4 Boundaries and Boundedness

These are two quite unrelated concepts which we will use frequently.

Boundedness first. A subset S of ² (open or not) is said to be bounded if and only if it is contained in some disc D(z0; r) of finite radius r; see Figure 1.8. The point is that a bounded set does not escape to infinity. Examples of bounded sets are any open disc, any closed disc, a single point, any finite set of points, and (of course) the empty set . The plane ², the upper half-plane, the x-axis are each unbounded.

Figure 1.8

The bounded domains (the open disc again) are an important subfamily of the family of all domains.

Now let us discuss the notion of the boundary of a set. Let S be any subset of ². A point z of ² is a boundary point of S if and only if every open disc D(z; r) centered at z contains some points in S and also some points not in S. Note that we do not require a boundary point of S to be an element of S. In fact, if S is itself an open disc, then its boundary points are precisely those on the rim of the disc, and none of these is a member of the disc.

It is worthwhile noting that, given a set S, a point z is a boundary point of S if and only if it is a boundary point of ² − S, the complement of S. A brief meditation should convince you that this is a reasonable property for a boundary point to possess.

Figure 1.9

Finally, we define the boundary (or frontier) of a set S to be the collections of all boundary points of S. The boundary of S is denoted S. See Figure 1.9.

Examples of Boundaries

1. Let S = D(z0; r), an open disc as usual. Then S is the circle of points z satisfying |z − z0| = r.

2. If = (z0; r), the closed disc, then is the same set as S in Example 1, namely, the circle of radius r centered at z0.

3. If Ω is the punctured disc D(z0; r) − {z0}, then Ω consists of the circle |z − z0| = r together with the point z0.

4. The boundary of ² is empty.

5. The boundary of the upper half-plane (y > 0) is the x-axis (y = 0). This is an example of an unbounded set with a nonempty boundary. Don’t confuse the two notions.

Preview

Our model of a nice bounded domain is the open disc D(z0; r). Its boundary is a very nice curve, a circle. In Section 1.2 we continue our study of domains by studying curves in the plane. In our applications, these curves will almost always arise as the boundaries of certain domains. Once we have completed our study of these boundary curves, we will begin at last to discuss the functions that live on our domains.

Exercises to Paragraph 1.1.4

1. Which of the following sets are bounded?

(a) |z| ≥ 1.

(b) A subset of a bounded set.

(c) 0 < |z − z0| < 1.

(d) The graph of y = sin x.

2. Determine the boundaries of the following sets. As usual, z = (x, y).

(a) x > 0, y > 0.

(b) |z − z0| ≤ 2.

(c) 0 < |z − z0| < 2.

(d) 0 < x < 1, y arbitrary.

3. Prove that a plane set S is bounded if and only if its closure (that is, S together with its boundary S) is bounded also.

Section 1.2 PLANE CURVES

1.2.0 Introduction

Curves—we know them when we see them, and yet to get an adequate terminology is surprisingly troublesome. First, therefore, let us look at the most important example, the circle, in some detail. This should make us more willing to accept the technical definitions to follow.

We let 0 = (0, 0) denote the origin of ² as usual, and C = C(0; r) be the circle of radius r > 0 centered at the origin, that is, the set of points z satisfying |z| = r. So far, C is a static set of points. Now we parametrize C as follows:

Let [0, 2 ] denote the interval of real numbers t satisfying 0 ≤ t ≤ 2 . Let = (t) be the function that assigns to each t in [0, 2 ] the point (t) of ², given by

We note first that | (t)| = r (recall sin² t + cos² t = 1) so that each point (t) does in fact lie on the circle C(0; r). We indicate this last statement briefly by writing

We can say even more. As the real number t increases from t = 0 to t = 2 , the point (t) travels once around the circle in a counterclockwise direction. See Figure 1.10. Note for instance that (0) = (r, 0), the starting point, and then

and we’re back where we started. Note also that the number t is the angle (in radians) between (t) and the x-axis.

Now things are no longer static. The function has imposed a direction of travel (counterclockwise) around the circle. We emphasize this by differentiating with respect to its variable t. Let us write

Figure 1.10

so that 1(t) = r cos t, 2(t) = r sin t (the coordinate functions for ). We differentiate (t) by differentiating its coordinates:

that is,

The pointed brackets here remind us that ′(t) is to be regarded as a vector (arrow), the velocity vector of . This is usually depicted with tail end at the point (t); see Figure 1.11. The velocity vector points in the direction of motion of the curve at the point (t). In the particular case t = /2, for instance, we have

Figure 1.11

Thus the arrow that we affix to the point ( /2) = (0, r) points r units to the left (since −r < 0) and zero units up or down. That is, (t) is moving directly to the left (counterclockwise!) at the instant t = /2.

Caution: There are many ways to run around a circle. For example, let : [0, 2 ] → C(0; r) be given by

You should convince yourself that as t increases from 0 to 2 , the point (t) travels three times around the circle in a counterclockwise direction. This new parametrization is essentially different from , even though the point set C(0; r) is the same in both cases.

Here is a clockwise parametrization of C(0; r). Let : [0, 1] → C(0; r) be given by

Then (0) = (0, r) is the starting point in this parametrization. By locating the points , you should convince yourself that (t) traverses C(0; r) once in a clockwise manner. Note also that was defined on the interval [0, 1], not on [0, 2 ].

1.2.1 Piecewise-smooth Curves

Now we make our definitions in the spirit of the preceding examples.

Let be a subset of ² and [a, b] an interval of real numbers t, a ≤ t ≤ b. (Note: = Greek capital gamma.) A function : [a, b] → , (t) = ( 1(t), 2(t)), is called a parametrization of if and only if

(i) is continuous; that is, 1(t), 2(t) are continuous functions of t; and

(ii) maps [a, b] onto ; that is, each z is of the form z = (t) for at least one t [a, b]. The variable t is called the parameter.

It can be seen that this definition of parametrization is too general for our purposes. There are too many continuous functions! Hence, we single out a particularly well-behaved class of parametrizations. We say that the parametrization : [a, b] → is smooth or continuously differentiable if and only if three further conditions hold, namely:

(iii) the coordinate functions 1(t), 2(t) are smooth in the sense that both derivative functions ′1(t), ′2(t) exist and are continuous for all t [a, b];

(iv) for each t [a, b], the velocity vector ′(t), defined as ′1(t), ′2(t) is different from the zero vector 0, 0 ;

(v) if, moreover, (a) = (b) (the curve is a closed loop), then ′(a) = ′(b) as well.

Figure 1.12

Let us discuss this definition; see Figure 1.12. Condition (iii) assures us that the velocity vector exists and depends continuously on the parameter t. We remark also that, by the derivatives ′1(a), ′2(a), ′1(b), ′2(b) at the end points t = a and t = b, we mean one-sided derivatives only. For instance,

Condition (iv) may be interpreted as follows: If we regard (t) as a point moving along , then its instantaneous direction is pointed out by the velocity (or tangent) vector ′(t). The vanishing of this vector—say, ′(t1) = 0, 0 —would mean that the moving point (t) stops when t = t1. It simplifies things greatly if we rule out this possibility.

We saw above three examples of smooth parametrizations of the circle. Note that each of these examples satisfies condition (iii) because sines and cosines can be differentiated again and again. How would you prove that the parametrization (t) = (r cos t, r sin t) given in Paragraph 1.2.0 satisfies condition (iv)? Hint: Use the Pythagorean theorem sin² t + cos² t = 1 to derive a smashing contradiction from the assumption ′ (t) = 0, 0 .

The most important plane curves for our purposes are the circle and straight line. You will find these treated at some length in the exercises.

Here are some more useful notions. The parametrization is simple if and only if the function restricted to the open interval (a, b)—that is, for t satisfying a < t < b—is one-to-one. In other words, if t1 and t2 are strictly between a and b and if (t1) = (t2), then t1 = t2. Geometrically, this means that the curve doesn’t cross itself, except perhaps at the end points. If the end points are equal (that is, if (a) = (b)) then we say that is closed, or a loop. See Figure 1.13.

Figure 1.13

The parametrization (t) = (r cos t, r sin t) for 0 ≤ t ≤ 2 is a simple closed smooth parametrization of the circle C(0; r). On the other hand, (t) = (r cos 3t, r sin 3t) for 0 ≤ t ≤ 2 is not simple because each point on the circle corresponds to three values of t (except for the starting point (r, 0) which corresponds to four values).

Actually, smooth parametrizations are not quite general enough. We wish to allow curves with a finite number of corners such as triangles and rectangles. At a corner, of course, we would not expect a unique direction or velocity vector. Again let : [a, b] → be a parametrization, (t) = ( 1(t), 2(t)). We say that is a piecewise-smooth parametrization of if and only if there exists a finite set of values a = a0 < a1 < a2 < < an = b such that the function restricted to the intervals [a0, a1], [a1, a2],…, [an−1, an] gives in each case a smooth parametrization of the subsets 0, 1,…, n−1 of defined by k = { (t)| t [ak, ak + 1]}.

Thus, a piecewise-smooth parametrization is one built up from smooth parametrizations joined end to end. In particular, a smooth parametrization is piecewise-smooth (let n = 1 in the definition above).

Example

We will parametrize the right triangle with vertices at the points (0, 0), (0, 1), (1,1); see Figure 1.14. It is simply a matter of building in three parts. Thus, let a: [0, 3] → as follows:

Figure 1.14

You should let t increase from 0 to 3 and check that (t) travels once around the triangle in a counterclockwise direction. Now we examine a corner point, say, (2) = (1, 1). By differentiating (t) = (1, t − 1), we see that the velocity vector ′(2−) in which t approaches 2 from the left (t < 2) is given by ′(2−) = 0, 1 . This points upward; the point (t) is climbing in the y-direction as t increases from t = 1 to t = 2. Likewise, by differentiating (t) = (3 − t, 3 − t), we compute ′(2+) = −1, −1 . This vector, with tail end fixed at the corner (2) = (1, 1), points toward the origin (0, 0) as expected.

Comments

1. We have not yet defined curve. Let’s attend to this now. Suppose : [a, b] → is a piecewise-smooth parametrization of the set . Then the pair ( , ) is termed a piecewise-smooth curve. However, it is common practice to speak of the curve , omitting mention of , or "the curve z = (t)." We will use both terms.

2. We are claiming that the set is a curve if it has a piecewise-smooth parametrization (roughly, a velocity vector). This corresponds to our intuition. Things are more delicate than you may imagine, however. It is possible to find a continuous : [a, b] → , (t) = ( 1(t), 2(t)), where is the two-dimensional unit square! A space-filling curve!