To go to a particular lecture, click on its number: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Where reading is indicated, you're advised to do it before the lecture, or at least give it a shot and be ready to ask questions.
You will find lists of problems labeled (O), (A), and (B). You should do each of the (O) and (A) problems on your own, or at least make sure you understand how to do them--whether or not you write them up to turn in. You are responsible for learning how to do such problems. The (B) problems are advisable to do but optional. In each assignment, you will be given a number of problems from lists (A) and (B) to write up and turn in by a certain date. The (O) problems cannot be turned in for credit. As you will see, you will have some choice as to how many problems you do, and which ones you do. If a problem is listed twice, it will count as two problems.
Please let me know if you find errors in the textbook, and about anything that you find unclear. Before you start using the book, consult the appropriate lists of corrections. If you have the second or later printing of the book, consult this list. If you have the first printing, consult also this list.
When you turn in a problem set, please fold each paper vertically, with your name and the date on the right-hand side of the back, and with your last name in the upper right-hand corner of every sheet. Label each problem with its section number and the problem number. For example, problem 6 in Section 1.2 should be labelled 1.2-6.
On the internet: By doing a search on the internet for "Complex Analysis," you may find a good many websites, providing the syllabi of courses at various universities, as well as colorful and dynamic graphics. You may find it amusing to look at some of these. You may even find it useful after you have studied mappings! Here is a link to one site that is nice to look at, and that has links to some other sites.
For a collection of sample questions from old tests, go to http://www.math.lsu.edu/~mcgehee/4036/Samples.pdf. No claim is made that these are necessarily representative of what will be on tests this fall; guidelines for study will be provided.
Read Section 1.1, subsection A. You may safely skip the other parts of 1.1. However, I suggest that you take a look at Figures 1.1-9 and 1.1-10, which show fluid flows which we'll study by means of conformal mapping in Chapter 5. The same mathematical model can be used to treat stress fractures in strength of materials.
Read Section 1.2. Figures 1.2-4 and 1.2-5 illustrate the mean value property which characterizes harmonic functions of two real variables.
Read Section 1.3. The material in Section 1.3 is based on familiar things: polar coordinates, two-by-two matrices, determinants. Know what's meant by "an argument." Understand the correspondence between two-by-two matrices and linear maps from the plane to the plane. Understand the properties, presented here, that such maps may or may not have, and how those properties are related, one to another. Linear maps from the plane to the plane may or may not be magnifications, pure magnifications, isogonal maps, rotations, conformal maps, etc.
It is important, both for concepts and for applications, to be able to understand mappings, particularly maps from the plane to the plane. Your objectives: Get in the habit of using precisely the terminology and the notation for images and pre-images. Practice visualization techniques.
Review: If you need to do so, review the basics of partial differentiation. Your calculus text or notes will have what you need. In this course, I want you to begin seeing equations involving partial derivatives as statements with geometric significance, not just as piles of symbols.
Problem Set 1, Due Thursday, September 2: Write up and turn in no more than eight of the (A) and (B) problems.
Solutions to most of these problems may be found in the papers by Jay Call and Jakob Duran. This URL will be available after Problem Set 1 has been turned in.
I'll continue to talk about the material of Sections 1.1-1.3.
Read Section 1.4. Your objectives are (1) to learn what an open set is; (2) to learn several other related definitions, like those of the words boundary, closure, closed, and interior point; (3) to understand how continuity can be defined in terms of open sets; (4) to understand connected sets; (5) to know the meaning of an open mapping; (6) why a non-constant open mapping on a connected domain cannot attain its maximum; and (7) given a harmonic function u(x,y), how to find a harmonic conjugate for it, v(x,y).
Read Section 1.5, which may be mostly a matter of review. The topics covered improper integrals, and the divergence, gradient, and potential of a vector field. Pay particular attention to Proposition 1.5.4. Probably new to you will be the definition of the Cauchy Principal value of an integral.
Lecture Preview: Differentiability, in the real-variables sense, of a function from the plane to the plane, means approximability locally, at each point, by a real-linear map. Real-linear maps from the plane to the plane correspond to 2-by-2 matrices. I'll discuss the special properties that such maps and matrices can have, following Section 1.3, and I'll talk also about the geometric meaning of Laplace's equation (Section 1.6).
To see movies that illustrate mappings from the plane to the plane, try this link, which will get you to one site that is nice to look at, and that has links to some other sites.
Read Section 1.6, A and B only. Also read Section 2.1.
Lecture Preview. Complex arithmetic should become second nature to you. By starting with the polar-coordinates definition of multiplication, I will try to convey the geometric meaning that enlivens complex-arithmetic computations like multiplication, division, conjugation, taking powers, and finding roots (for example, the 8th roots of 2).
Look for the answers to these questions: What are the linear maps from the complex plane to the complex plane? How are they different from the linear maps from the real plane to the real plane?
In subsection G, you'll find the complex forms for equations of lines and circles. As subsection H points out, knowing those forms makes it easier to understand how they are mapped by the reciprocal map and by the mapping J, which is reflection in the unit circle. What we find is that either map takes every circle and line to a circle or a line. Notice that G is the composition of reflection in the unit circle, J, with reflection in the real axis, conjugation. We'll need these results in Section 5.2.
Important: Please let me know if you find errors in the textbook, and about anything that you find unclear. Before you start using the book, consult the appropriate lists of corrections, and pencil the corrections into your copy of the book, to prevent getting hung up on an error or mispring. If you have the second or later printing of the book, consult this list. If you have the first printing, consult also this list.
Problem Set 2, Due Thursday, September 9: Write up and turn in no more than eight of the (A) and (B) problems. Note: The definition of "neighborhood" appears in Exercise 21 on page 60.
Solutions to some of these problems may be found in the papers by Jakob Duran and Miao Xu. This URL will be accessible after the papers are turned in.
Lecture Preview. I'll begin to discuss with care some of the important basic mappings, starting with the exponential and the logarithms. There are many logarithms, as there are many arguments, at each point. The question of when a set has an argument defined on it is essentially the same question as to when it has a logarithm defined on it.
The mapping properties of the sine function are worth understanding thoroughly. We will see its usefulness in certain boundary value problems. As you can tell by looking at the defining expression for the sine, horizontal and vertical lines make a good grid to work with in the domain.
Lecture Preview. I'll discuss the boundary value problems of the kind which we will consider in Chapter 5, and present the basics of the conformal mapping method. The importance of this method is our motivation for the careful study of mappings. I'll discuss the sine mapping in detail and present some temperature distribution problems and their solutions.
Read Section 2.3. The condition of approximability by a linear function unifies all the definitions of differentiability. The meaning of "a linear function from the plane to the plane" depends on whether the real plane or the complex plane is meant. The complex plane possesses more algebraic structure--it's a field--so linearity becomes a more restrictive requirement.
Recall that a linear function from the real plane to the real plane is given by a 2-by-2 matrix (the function is evaluated at a 2-vector by multiplying it by the matrix on the left). A linear function from the complex plane to the complex plane is given by multiplication by a complex number. (If we represent it in real notation, then the 2-by-2 matrix is of a quite special kind).
As given in Section 2.3, the definition of a holomorphic function f(x+iy)=u(x,y)+iv(x,y) on an open set entails, at each point, the existence of the four first-order partial derivatives (of u and v with respect to x and y) and their obedience to the Cauchy-Riemann equations, plus the condition of the good linear approximation. It may appear to you at this point that holomorphy does not imply the continuity of the four first-order partials, or even the existence of the second-order partials, much less the harmonicity of u and v. But in fact, it does imply those things and much more.
Read Section 2.4. The important topics here, some of which I'll discuss in lecture, are as follows: Sequences and subsequences. For real-valued sets or sequences: upper bounds, least upper bounds, limit superiors. Continuity and closed sets characterized in terms of sequences. Compactness. Continuous functions preserve compactness. Uses of compactness. Sequences of functions. Uniform convergence.
Read Section 2.5, subsections ABC.
Lecture Preview. More on sequences. Curves. The significance of the "smooth" requirement. Curves and paths, defined. The integral of a complex-valued function along a curve or a path. Evaluation by the definition, that is, using a pullback. Integrals along line segments and around circles.
Problem Set 3, Due Thursday, September 23: Write up and turn in no more than eight of the (A) and (B) problems. Note: The definition of "neighborhood" appears in Exercise 21 on page 60.
Solutions to some of the problems in Problem Sets 2 and 3, due to Andrew Morrow and Mohammad Almutawa, are posted at http://www.math.lsu.edu/~mcgehee/4036/ps2-3.pdf
Read Section 2.5 DEFGHI. Lecture topics: Pathwise connected sets, independence of path, Morera's Theorem (2.5.12). A review of arguments and logarithms. An important example of a holomorphic function (on a punctured plane) whose integrals are not independent of path. Winding numbers: techniques, examples.
Bring questions and problems you want to see done in class. I'll offer some advice about the test on Thursday.
I recommend reading all of Section 2.5 with care, including the proof of Goursat's Lemma and the interpretation of the real and imaginary parts of a complex integral around a closed path as circulation and flux of a fluid flow, and the Cauchy-Riemann equations' relation to irrotational and incompressible fluid flows. I'll talk about this material today as time allows. Problems to do on your own:
The test will cover all the material through Tuesday's lecture. Study all the reading assigned above, all the problems on lists (O) and (A), and all the lectures. This will be an 80-minute test. It will be closed-book, closed-notes. No calculators, computers, or cellphones allowed.
At http://www.math.lsu.edu/~mcgehee/4036/Samples.pdf you will find four pages of Sample Test Problems. Although these don't represent a complete study guide, you may find it helpful to look at the following problems:
A copy of Test 1, given 9/30/04, is at http://www.math.lsu.edu/~mcgehee/4036/T1.pdf. A key to Test 1 is at http://www.math.lsu.edu/~mcgehee/4036/T1Key.pdf.
Read Section 2.6. In your reading, pay special attention to how you can find the radius of convergence of a series, by applying the Cauchy-Hadamard Theorem (p. 172) and by other methods; and to methods of finding the Taylor series of a given function.
Lecture Preview: More on integrals. Power series. Disk of convergence, radius of convergence; statement of the Cauchy-Hadamard Theorem. Subuniform convergence on the disk of convergence. Finding the radii of convergence. Starting with a power series that converges to a sum f(z) for z in the disk of convergence, we define the derived series in a formal way. We prove that its radius of convergence is the same as that of the original series, and that it converges to the derivative f'(z). Then we can show that the power series representation for f is unique; it must be the Taylor series for f. The geometric series. Manipulating the geometric series to produce power series representations of certain functions.
Reminder: Make systematic use of the lists of corrections to the book, perhaps penciling in the necessary changes to your copy of the book. If you have the second or later printing of the book, please consult this list of corrections and other emendations. If you have the first printing, here is a further list of corrections.
Lecture Preview: I'll start with a review of major results that we've covered, beginning with Goursat's Lemma, and proceed into Chapter 3 and the Cauchy Theory, including the proof that a holomorphic function has (1) an integral represenation and (2) a power series expansion.
Problem Set 4, due Thursday, October 14: Write up and turn in at most 9 of the (A) and (B) problems.
Read Sections 3.1 and 3.2. Read about the Fundamental Theorem of Algebra, which is proved here using Liouville's Theorem (a bounded entire function is constant). The objective is to tie together the conditions of (1) pointwise differentiability in the complex sense; independence of path for integrals; representability by power series; and representability by a Cauchy integral. We're going to move fairly rapidly to obtain what we need from Chapter 3, and then enter Chapter 4, where the theme is the easy evaluation of complex integrals. You'll begin to see the techniques right away.
Lecture Preview. Examples using Cauchy's Theorem for a Convex Set. If the zeros of a holomorphic function on a connected set have an accumulation point within the set, then the function is identically zero. Cauchy's Theorem on an arbitrary open set O, stated and proved in Section 3.2, is quite powerful and easy to use. In this version of Cauchy's Theorem, the condition that O be convex is replaced by the condition that the contour have winding number zero with respect to every point outside O ("the contour is homologous to zero in O"). I'll offer examples of contours that are homologous to zero, and why it's nice to use one that goes around a large part of O. One application is the development of the Laurent Series Representation for a function that is holomorphic on an annulus.
I leave it to you to study the proof of Cauchy's Theorem that appears in Section 3.2 if you like. There are two additional proofs in Chapter 6. You should know the statement of the Theorem, and the definition of a simply connected set.
Lecture Summary. Examples in which a Laurent series is obtained. Isolated singularities. Classification: Removable singularities, poles, and essential singularities. Definition of the residue. Finding residues.
Problem Set 5, due Thursday, October 21: Write up and turn in at least 4 and at most 10 of the (A) and (B) problems. If you do 1, 2, or 6, make a sketch, and explain clearly how you use the Theorem to obtain your answer. (Note that the answers are given.)
After October 21, solutions by Jay Call and Jakob Duran to some of these problems will be available at http://www.math.lsu.edu/~mcgehee/4036/ps5.pdf. More solutions are at http://www.math.lsu.edu/~mcgehee/4036/ps5-2.pdf.
Lecture Summary More examples in which we classify singularities and evaluate residues. Once we understand isolated singularities, we can see the Residue Theorem as a staightforward consequence of Cauchy's Theorem. I'll present some basic examples of how to use the Residue Theorem to evaluate integrals of rational functions over the line (Section 4.3).
Read Section 3.3, subsections ABCD; also Sections 4.1 and 4.3.
Lecture Topics: An isolated singularity at infinity. More applications of the Residue Theorem and its applications. I'll discuss the Residue Calculus methods of Section 4.1, in which the integral of a rational function in the sine and cosine is recognized as the pullback of a complex integral around the circle, and the Residue Theorem is applied to find the value of the integral. A holomorphic function on a simply connected open set that has no zeros thereon "has a logarithm" and "has a square root." See 3.2B.
Problem Set 6, Due Thursday, October 28: Do at least 4 but no more than 10 of the (A) and (B) problems.
Solutions to some of the problems appear in this work by Evan Anzalone, Jay Call, and Jakob Duran, http://www.math.lsu.edu/~mcgehee/4036/ps6.pdf, which will be accessible after October 28.
Read Section 3.4E, especially pages 244-245.
Lecture Topics: Preview of the Argument Principle. More applications of the Residue Theorem.
To prepare for Test 2 on Thursday, November, 4, study all the material of the course covered so far, because even the basics from the early weeks are likely to be involved. You should expect the problems to reflect material from Lecture 8 forward. The test will ask you to find values for integrals along curves, using a variety of methods - pullbacks, antiderivatives, Cauchy's Theorem, and the Residue Theorem. You may also expect problems on series.
Among the Sample Test Problems, for which the link appears above (near the top, above "Lecture 1"), pay special attention to these: In "Test 2 Problems," numbers 1, 5,.6, 7, 8; "Test 3 Problems," problem 1; and in the "Final Exam Problems," problems 2 and 3. Also, see problems 3-7 on Test 2 from spring of 2004 at http://www.math.lsu.edu/~mcgehee/4036/T2-04.pdf and the notes on that test at http://www.math.lsu.edu/~mcgehee/4036/T2-04Notes.pdf.
Read Section 3.4, subsection E, especially pages 244-255; and Section 4.4, especially Examples 4.4.2 and 4.4.7. Example 4.4.7 evaluates the Fresnel integrals. Both 4.4.2 and 4.4.7 involve the exponential function, but not the Residue Theorem.
Lecture Summary: The Casorati-Weierstrass Theorem. Review for Test 2.
As of November 8, Test 2 is posted at http://www.math.lsu.edu/~mcgehee/4036/T2.pdf, and a key is posted at http://www.math.lsu.edu/~mcgehee/4036/T2Key.pdf.
Lecture Summary: I will evaluate the Fresnel integrals, which requires a delicate estimation of certain integrals on a semicular arc; see Examples 4.4.7 and 4.4.2. I'll also discuss the Argument Principle and show a movie showing the image of a circle centered at the origin, as its radius grows, under a polynomial mapping. Section 3.5, which is optional reading, shows you some "still photographs" of the same type.
Read Section 5.1 and review the earlier lectures in which I discussed the conformal mapping method in general and showed you how to solve some "easy" boundary value problems, and presented some of the other problems in this Section. By an easy problem, I mean one in which we can easily pick out a solution from the list of harmonic functions that we are familiar with. I'll discuss the conformal mapping method in general and talk about examples like those in Section 5.1. Pay special attention to the boundary value problems in which the domain is the upper half-plane, so that the boundary is the real axis; and in which the boundary-value function takes on a finite number of values, each on an interval of the real axis. The method, the linear combination of arguments, is "easy" in the sense described above, at least after it's once pointed out to you.Problem Set 7, Due Thursday, November 11: Do at least four but no more than ten of the problems from the (A) and (B) lists.
Here are some solutions due to Ashley Long, at http://www.math.lsu.edu/~mcgehee/4036/ps7.pdf.
Read Sections 5.1 and 5.2.
A note about Section 5.1: Think about the Example illustrated in Figure 5.1-2. Rather than just produce expressions for a solution, take the time to understand what the solution means physically in, for example, a heat flow problem. In this problem, for example: What's the limit of the temperature along each vertical half-line? What's the limit of the temperature along each horizontal half-line to the left? --to the right? Observe that the 50-degree isotherm is the only one that meets the x-axis at a point other than one of the two insulated points. Observe the behavior of the isotherms for temperatures greater than/less than 50 degrees; draw, for example, the 49-degree isotherm.
It's clear that a larger collection of conformal mappings would be useful. It is quite worthwhile to become an expert on Mobius maps. They are the only mappings which are conformal and one-to-one from the extended plane to itself. To become familiar with them, learn the logic of their basic properties. For example: Mobius maps preserve Circles. A Mobius map, when its values are known at three points, is completely determined.
Read and study Section 5.2 thoroughly, and get started early on the problems in that Section. About 25 percent of the final exam will deal with conformal mapping methods. It will be worthwhile to become expert on Mobius maps. One objective is to learn how, given a Mobius map, to understand "what it does" - what regions map to what regions.
Lecture Summary: More on Mobius maps.
Problem Set 8, due Thursday, November 18: Do at least four but no more than ten of these problems:
Solutions by Miao Xu, Abhilash Ponnath, Yifan Qiu, and Damianos Christofides posted at http://www.math.lsu.edu/~mcgehee/4036/ps8.pdf.
Lecture Summary: Today I considered a temperature distribution problem on a crescent, using a Mobius map and a power map to make the crescent conformally equivalent to the upper half plane. Compare Example 5.2.14 and Example 5.1.5. Some Mathematica graphics illustrating the solution on the upper half-plane may be found at http://www.math.lsu.edu/~mcgehee/4036/temp-11-18.pdf. A more complete set of graphics, due to Jude Melancon, is available at http://www.math.lsu.edu/~mcgehee/4036/temp-Melancon.pdf.
Lecture Summary: More conformal mapping problems. Fluid flow problems.
Lecture Summary: More on fluid flow problems; more on Mobius maps. The Mobius maps of Example 5.2.18 are conformal equivalences of the unit disk with itself. Figure 5.2-4 shows a typical case. They have applications, for example, in electric-field problems, as in Examples 5.3.2 and 5.3.3.
Problem Set 9, due Thursday, December 2: Do at least four but no more than ten of these problems:
The department's scanning function is not working. If it is fixed before the final exam, I'll post some solutions for PS9, by Waheed Syed and Marshall Harper, at http://www.math.lsu.edu/~mcgehee/4036/ps9.pdf
Lecture Summary: More conformal mapping applications.
The Final Exam will be open-book, open-notes, no cell phones, no computers or calculators. It will be comprehensive, except that about 25% will be devoted to conformal mapping problems.