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Today we see the study of boundary behavior as properly part of complex function theory. Reproducing kernels, such as the Poisson and Cauchy Chapters 1 and 8 , are basic tools in the subject. So are harmonic measure Chapter 9 and conformal mapping Chapters 4 and 5. Conversely, results about the boundary behavior of holomorphic functions shed light on mapping theory and the construction of kernels.

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Thus there is a new wedding of real variable techniques with complex variable techniques. This has indeed proved to be a fecund meeting ground, and we shall indicate some of its features in the present chapter. The reader of this chapter will want to be comfortable with elementary hard analysis and estimates. What does this mean? In fact results of this sort are true for fairly large classes of holomorphic functions f. First, it is important to understand that without additional hypotheses on f , there is no hope.

Let us begin by just discussing smooth f —even real analytic f. The function 1 6. The Bagemihl—Seidel theorem, discussed below Section 6. By contrast, in the year —shortly after Lebesgue formulated his new theory of measure and the integral—Pierre Fatou proved the following remarkable theorem: Theorem: Let f be a bounded holomorphic function on the unit disk. He viewed the putative limit as Abel summation of the Fourier series for f. In common parlance, we say that Fatou proved that a bounded analytic function has radial boundary limits almost everywhere.

See Figure 6. Is the result true on a more general class of domains? The third question can be answered with the uniformization theorem, or with various localization techniques. We cannot treat them here. Let 0 6. Radial boundary limits. See Y. Proposition 6. Let 1 Proof. Remark 6. The function M f is called the Hardy—Littlewood maximal function of f. Lemma 6. The operator M is of weak type 1, 1. By Proposition 6.

A Stolz region. In the last estimate we used Lemma 6. Theorem 6. Why are holomorphic functions better? The next result of Jensen is crucial. Corollary 6. Apply Lemma 6. So Proposition 6. The conclusion that the limit exists follows from Theorem 6. Therefore, by Proposition 6. A fortiori, F has nontangential boundary limits almost everywhere. It follows that f does as well. For, by Theorems 6. In any event, we see that some extra care must be taken if we expect that the holomorphic functions we are studying will have boundary limits.

Much of modern harmonic analysis, as well as the function theory of the disk, is predicated on the interplay of Fourier analysis on the boundary and complex analysis on the interior that is facilitated by Theorem 6. It is a fundamental result. If we take Theorem 6. Let f be a bounded holomorphic function on the unit disk. Choose a subsethat converges uniformly on K. Call the limit function f0. In their seminal paper [LEV], Lehto and Virtanen extracted one of the key ideas from this last proof.

For example, any bounded holomorphic function is certainly normal. Any holomorphic function that omits two values is certainly normal. Any univalent function is normal. Let f be a normal holomorphic or meromorphic function on the unit disk. It is a deep insight of Lehto and Virtanen [LEV] that the hypotheses of these two results may be considerably weakened.

In fact we may assume that f has a limit along any curve—even a curve in the boundary. And the same conclusion follows. We shall present some of these ideas now. A good reference for further reading is [GOL]. We leave it to the reader to translate the ideas over to the disk by way of the Cayley transform. We begin with an estimate for harmonic measure. Proof of the Lemma. This establishes inequality 6. The statement about equality is just the maximum principle. We leave the details to the reader, or see [GOL, pp. Of course this is just the maximum principle.

Furthermore, u is equal to 0 on A and is nonnegative on B. Let f be holomorphic and bounded on the upper halfplane U. We shall continue to use notation and normalizations from the previous proposition. Applying inequality 6. Prove the failure of this assertion. Prove it. Let f be a holomorphic function on the disk. Problems for Study and Exploration Prove that a function in the Nevanlinna class has radial boundary limits almost everywhere. Refer to Exercise 4 for terminology. This problem is philosophically related to the preceding one.

The classical result, once again, is due to G. Hardy and J. Littlewood [GOL]. A good reference for this sort of result is I. In Proposition 6. In its earliest days, one of the chief teclmiques was power series. During the nineteenth century, thanks t. Today, when we speak of "classical function theory," we are usually thinking of those eighteenth- and nineteenth-century techniques. TWs approach usua lly involves a combination of estimates and qualitative reasoning.

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In the late nineteenth and in the twentieth century, a variety of powerful new tools were developed. These include geometric methods, algebrn. T he present collection of six chapters will concentrate on classical function theory and geometry. More modern techniques will be explored in later parts of the book. By then the reader will be inured to the idea of adopting new points of view, and will perhaps be a bit more adventurous. Less familiar to most students, and an idea that has come into its own only in the past forty years, is the idea of studying the inhomogeneous Cauchy—Riemann equations.

In the present chapter we explore this circle of ideas.

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Later in the book, we shall see equation 7. A good understanding of multivariable calculus is all that is required for an appreciation and understanding of the ideas in this chapter. The Cauchy—Riemann equations result. The equation has considerable intrinsic interest, but it also can be useful for constructing holomorphic functions. Then we have this standard result from calculus: Theorem 7.

Note that it is valid for essentially all functions—not just holomorphic functions. Corollary 7. Putting all this information into equation 7. With hypotheses as in Corollary 7. Now we derive an explicit integral solution formula for the inhomogeneous Cauchy—Riemann equations. We will see that the holomorphicity of the onedimensional Cauchy kernel will play an important role in the derivation of this formula. This is the result that we wish to prove. The result for higher k follows by a simple induction. The proof is therefore complete. Remark 7. One can make an even sharper statement if one uses precise function spaces that have been designed for this purpose.

More information on these matters may be found in [KRA2]. Then any solution v of the equation 7. Let u be the solution constructed in Theorem 7. We proved in Proposition 7. Now an important point must be made. In fact we need to know how to solve the equation 7. The next theorem addresses this need: 7. Now, using Corollary 7. Is this solution unique? Repeat Exercise 3 with z on the right-hand side of the equation replaced by z. Problems for Study and Exploration 5. If f is holomorphic and nonvanishing, then verify that log f is harmonic.

We shall explore all these additional ideas later in the book. But we will also make good use of these results elsewhere in the book, particularly in our study of conformal mappings and the corona problem. This is the Dirichlet problem. For us, the Dirichlet problem is a useful device for constructing harmonic functions. Perron has given us a method for solving the Dirichlet problem. A typical theorem will be stated in a moment. See Figure 8. Theorem 8. A barrier. Just use the implicit function theorem. In fact for many domains there is another approach to the Dirichlet problem that will be of particular interest for us.

On the disk—which is a very special domain indeed—there are ad hoc methods for deriving the Poisson kernel. Let us outline one of these now. It will be convenient in these discussions to momentarily forget the complex structure of C and instead think of ourselves as living in R2. Proposition 8. Remark 8. This result is so important that it merits discussion. Lemma 8. The details of the proof may be found in [KRA1, Chapter 1]. Corollary 8. There are several useful ways to restate the maximum principle.

This statement has a strong intuitive appeal. Proof of the Corollary. Prove the following trivial variant of Theorem 8. The reader should check that Properties 1 — 4 uniquely determine the function G. We also have not said much about the dependence of G on the x variable. A now familiar computation see the proof of Proposition 8. A limiting argument then completes the proof.

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Details are left to the interested reader. However, we can compute it for the disk. Elsewhere in the present book we have indicated how to determine the Poisson kernel for the disk by summing a Fourier series. Now we calculate the kernel using the paradigm of Theorem 8. Exercise for the reader just compute. It may now be checked directly that in fact Fx is smooth at 0, so it follows from the continuity of the derivative that Fx is harmonic on a neighborhood of D.

The expression in polar coordinates is an immediate consequence. Inequality 1 is obvious. We already know that the Poisson kernel is harmonic in the x variable by Corollary 8. Since F is obviously continuous indeed smooth 8. This is what we wished to prove. Exercises for the Reader 1. Give another proof of Theorem 8. Give a proof of Theorem 8. Examine the proof of Theorem 8. Hence, by the Hopf lemma see Lemma 5.

From this, Theorem 8. Provide the details of the proof of Proposition 8. This exercise outlines a method to construct the Poisson kernel that is analogous to the Bergman kernel construction, by way of a basis, in Chapter 2. Let us consider the L2 functions on the unit circle of the complex plane that are boundary functions of holomorphic functions on the disk. Use the Hopf lemma to derive a version of the maximum principle for harmonic functions. Without some regularity hypothesis on the boundary of the domain, the Dirichlet problem may fail to have a solution.

It has become an essential tool in potential theory and in studying the corona problem. It is useful in studying the boundary behavior of conformal mappings, and it tells us a great deal about the boundary behavior of holomorphic functions and solutions of the Dirichlet problem. All these are topics that will be touched on in the present book. From the point of view of function algebras which we discuss in Chapters 4 and 11 , harmonic measure may be thought of as a representing measure for a multiplicative linear functional.

It is a device that bridges the techniques of hard analysis, measure theory, functional analysis, and operator algebras. A good grounding in real analysis will be helpful for an appreciation of the ideas in this chapter. We now review some of the key elementary ideas connected with this fundamental equation. We quickly review its essential features. Thus u is to be the harmonic continuation of f to the interior of U. The Dirichlet problem has both mathematical and physical interest.

If U is a thin metal plate, formed of heat-conducting material, and if f represents an initial heat distribution on the boundary of the plate, then the solution u of the Dirichlet problem represents the steady-state heat distribution in the interior. We have treated the Dirichlet problem in detail in the last chapter. Thus we should be able to obtain the corresponding solution of Dirichlet problem. Then the corresponding solution of the Dirichlet problem should be 9.

For a general domain, one can almost never know the Poisson kernel explicitly. Proposition 9. It is important also to observe that the Dirichlet problem may be solved for more general boundary data than a continuous function. This fact will be crucial in what follows. Such a Dirichlet problem is solved by the Perron method.

If it does exist, is it unique? Let us see why. It is a classical result see [AHL2] as well as our Section 4. Remark 9. Furthermore, by classical results coming from potential theory see [AHL2] , any domain on which the Dirichlet problem is solvable will satisfy the hypotheses of the theorem. We will remove this extra hypothesis later.

Let the exceptional boundary points at which 9. These may not be domains i. Therefore, by the usual maximum principle, u is identically equal to the constant N. But then the boundary condition 9. Therefore 0 9 Harmonic Measure Proposition 9. It is worth recording here an important result of F. Riesz see [KOO, p. See also Theorem 9. It has become a powerful analytic tool. Example 9. Glancing at Figure 9. But we can say more. See Figure 9. The angle subtended by E at z. Now it is helpful to further rewrite this last expression again using 9.

It has proved useful in various parts of analysis, notably in proving the Riesz—Thorin interpolation theorem for linear operators. Here we shall give a thorough treatment of the three circles theorem from the point of view of harmonic measure. Afterwards we shall discuss the result of Riesz and Thorin. We begin by treating some general comparison principles regarding the harmonic measure. Often this standing hypothesis will go unspoken.

Of course we consider only holomorphic f because we want a mapping that preserves harmonic functions under composition. The result is now immediate. And of course it is harmonic. Theorem 9. Let A be a closed set. If f z 9 Harmonic Measure Remark 9. This result is of fundamental philosophical importance. It shows how the harmonic measure is a device for interpolating information about the function f.

Namely, we have this version of the three circles theorem. Then 9. It is proved from Theorem 9. In this text we shall concentrate on the complex method, which of course was inspired by the ideas of Riesz that were described at the beginning of this section. Instead of considering the general paradigm for complex interpolation, we shall concentrate on the special case that was of interest to Riesz and Thorin. We now formulate our main theorem. Fix a nonzero function f that is continuous and with compact support in R2.

We will prove an a priori inequality for this f , and then extend to general f at the end. Obviously this is grist for the three-lines theorem. This is the desired conclusion for a function f that is continuous with compact support. Riesz Theorem One of the classic results of function theory was proved by the brothers F. It makes an important statement about the absolute continuity of harmonic measure.

We begin with a preliminary result that captures the essence of the theorem. Riesz Theorem Theorem 9. Let 0 9. Riesz that we described in Remark 9. Proof of Theorem 9. The reason that we include Theorem 9. Let notation be as in Theorem 9. Since F is an arc, we see that Theorem 9.

We now summarize this result in a formally enunciated theorem. Riesz, We conclude by noting that Theorems 9. What can you conclude about E? What does harmonic measure—on the disk let us say—have to do with the Poisson kernel? Give a detailed answer. Refer to Exercise 5. Refer to Exercises 5 and 9. Let D be the disk. Refer to Exercise There are two basic types of convergence: pointwise convergence and norm convergence.

The study of norm convergence gives rise rather quickly to the study of the Hilbert transform. This statement alone was a triumph of the very new subject of functional analysis. It is straightforward to see that the Hilbert transform is bounded on L2. In the present chapter we will tell a good part of this story. There is also some functional analysis thrown into the mix. It is the stepping stone to a lot of good analysis, much of which is still studied today. As was proved in Section 6. The positivity of u implies that F is bounded.

By Theorem 6. Then Then the imaginary part of the Cauchy kernel, as just normalized and calculated, equals 1 r sin t. Its study is motivated by questions of norm convergence of classical Fourier series. This kernel is a transcendental function, and is tedious to handle.

But what gives us the right to replace the complicated integral In particular, it is in every Lp class.

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So we think of P. In practice, harmonic analysts study the integral in The Hilbert transform H is one of the most important linear operators in all of mathematical analysis. First, it is the key to all convergence questions for the partial sums of Fourier series. Second, it is a paradigm for all singular integral operators on Euclidean space.

Third, the analogue of the Hilbert transform on the line is uniquely determined by its invariance properties with respect to the groups that act naturally on 1-dimensional Euclidean space. Notice that this interval contains one element from each equivalence class in T. Let f be an integrable function on T. Let f be integrable. That proves the result for trigonometric polynomials. And he proved a version of our Theorem Let f be an integrable function on T and let the formal Fourier series of f be as in In other words, using It is the fundamental object in any study of the summation of Fourier series.

Contrast this result with the situation for Taylor series! Proof of the Theorem. Call it g t. The other three parts of I are handled in the same way. That takes care of I. The analysis of II is similar, but slightly more delicate. That is what we wished to prove. Here eN is the operator that shifts the Fourier series of f by N i.

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To understand this last equality, let us examine a piece of it. The other parts of We shall see that the Hilbert transform is bounded on L2. We shall give an independent proof below that H is bounded on Lp for 1 10 The Hilbert Transform they are Hilbert space projections. Taking these boundedness assertions for granted, we now re-examine equation Multiplication by a complex exponential does not change the size of an Lp function in technical language, it is an isometry of Lp.

So And the norm is plainly bounded independent of N. It is useful in the study of the Hilbert transform to be able to express it explicitly as an integral operator. The next lemma is of great utility in this regard. Lemma A rigorous proof of this lemma would involve a digression into distribution theory and the Schwartz kernel theorem. So this lemma will play a tacit role in our work. If we apply Lemma Instead we use Abel summation i. It should be noted that we suppressed various subtleties about the validity of Abel summation in this context, as well as issues concerning the fact that the kernel k is not integrable.

For the full story, consult [KAT]. Of course the connection is much more profound than that; much of the study of modern complex function theory, function algebras, and Fourier series hinges on this connection. We resolve the nonintegrability problem for the integral kernel k in Therefore the integral in We now reiterate the most fundamental fact about the Hilbert transform in the language of integral operators: Theorem It is bounded on Lp when 1 Norm-convergent partial summation of Fourier series is valid in the Lp topology if and only if the integral operator in An immediate corollary of Theorems Theorem The Hilbert transform is bounded on Lp , 1 We know that the Hilbert transform is bounded on L2 , L4 , L6 ,.

We may immediately apply the Riesz—Thorin theorem Section 9. That completes the proof. Remark And we may replace the Hilbert transform by any convolution operator. Details are left for the interested reader. Now let us resume the proof of Proposition The Hilbert transform is an isometry of L2. Explain this statement and prove it. But the kernel of the Hilbert transform is not integrable to any power. Prove that the Hilbert transform is a self-adjoint operator in a suitable sense. How does this result generalize that of Exercise 9?

See [STE] for further details.

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