Math 448: Complex Analysis

About Math 448

In Math 331 you saw how mathematicians were led to construct the real number system because certain elementary problems had "solutions" that were not rational numbers. For example, the length \(x\) of the diagonal of a unit square cannot be expressed as a rational number, since by the Pythagorean theorem, we must have \(x^2 - 2 = 0\). But you proved that \( \sqrt{2}\) is not rational. Such quantities, which arose in the most elementary physical way, could not be ignored, even if they could not be adequately described or defined. Thus some level of acceptance of irrational numbers existed even before Dedekind's work in the nineteenth century.

Even more fundamental for European mathematicians was the problem of negative numbers. It was not until the seventeenth century that negative numbers became known in Europe through Arab texts, though the Hindus had used them as early as the seventh century. Why were negative numbers problematic? Consider this argument given by Antoine Arnauld (1612--1694). We wish to claim that \(-1:1 = 1:-1\). Of course \(-1\) is less than \(1\). How can a smaller number be related to a larger one in the same way that a larger number is related to a smaller one?

Without quite understanding negative or irrational numbers, mathematicians began to struggle with what we now call complex or imaginary numbers. Again the motivation was a search for solutions. The quadratic equation \(x^2 - 2 = 0\) at least has irrational solutions, but the equally simple equation \(x^2 + 1 = 0\) has no real solutions, rational or irrational. This fact follows directly from the axiom system for real numbers which guarantees that for any real number \(x\), we have \(x^2\) is non-negative. But then \(x^2 + 1 > 0\). That is, no real number \(x\) satisfies \(x^2 + 1 = 0\).

As early as the sixteenth century mathematicians began to explore the properties of nonreal solutions to quadratic equations. The symbol \(i\) was used to denote the "imaginary" solutions of \(x^2 + 1 = 0\). Acting as if the usual algebraic laws were valid here, mathematicians such as Jerome Cardan (1501--1576) began to use square roots of negative numbers to solve all sorts of "impossible" problems. In his Ars Magna of 1545 Cardan poses the following problem. Find two numbers whose sum is \(10\) and whose product is \(40\). Of course, this requires us to find solutions to the quadratic equation \(x(10 - x) = 40 \), or equivalently, \(x^2 - 10x + 40 = 0\). Using elementary calculus, the quadratic \(x^2 - 10x + 40\) has an absolute minimum of \(15\) at \(x = 5\), as long as \(x\) is constrained to be a real number. Therefore, no real solutions to this problem exist. Cardan was not deterred, however, and he obtained the nonreal "solutions" that we today would denote by \(5 + \sqrt{-15}\) and \(5 - \sqrt{-15}\), that is, \(5 + \sqrt{15}i\) and \(5 - \sqrt{15}i\). You can verify that Cardan's answers are correct, acting on faith that the usual laws of algebra apply to such quantities.

Cardan and others at this time operated without any careful or formal theory of complex numbers by blindly applying the ordinary rules of algebra as needed. But some mathematicians completely rejected the notion of such "imaginary" solutions to equations. Felix Klein wrote:

Imaginary numbers made their way into arithmetic calculation without the approval, and even against the desires of individual mathematicians, and obtained wider circulation only gradually and to the extent they showed themselves useful. Imaginary numbers long retained a somewhat it mystic coloring.... As evidence, I mention a very significant utterance by Leibniz in the year 1702, "Imaginary numbers are a fine and wonderful refuge of the divine spirit, almost amphibian between being and non-being."

Our immediate goal is to make precise the nature of the complex number system and its rules of algebra (it is a field). This will be followed by developing a "calculus" that employs complex-valued functions of a complex variable. This process is analogous in many ways to the development of the calculus in Math 331, but with some interesting and beautiful differences. This work will require us to extend the notions of limit, continuity, differentiation, integrability, and convergence to functions of complex variables. Your previous work in Math 232 and Math 331 will be very useful here. Keep a copy of Foundations of Analysis handy.

Expectations and Assessment

In a 400-level class, the general expectation is that you will be focused, independent, learners. My job is to point you in the right direction and to explain some of the difficult results. To be successful in this course and, more importantly, to come to appreciate the beauty of complex analysis, will require substantial effort. I expect that you will spend four or five hours of work on your own for each hour of class time. We will make use of relevant material from previous courses, especially Math 331, Math 204, and Calculus I--III. We will not have time to cover every proof in the text, so carefully read the material and ask questions when you are confused.

Below, I outline the readings for the term. I hope that we can stick to this schedule, as the material on residues is a fitting place for the course to end. I will assign both practice and graded problems out of the text as well as problems from other sources.

The main component of your grade for the course will be a series of 12 to 15 graded homework assignments. While you may discuss this work with classmates, the work you hand in must be your own. The graded homework assignments will contain both routine and more challenging problems which extend the material we are discussing in class. Seek extra help from me when necesssary. I expect to have you present some of your work in class and I am likely to have you present some of the proofs of the theorems we will cover.

Since the class is small and this is a four-hundred level course, there will be two out-of-class self-scheduled exams. You will have 90 to 120 minutes to work on these exams at a time during the day when it is convenient for you. There will also be a final exam which may include a take-home portion or a seminar presentation.

Your course grade will be calculated as follows: Homework 45%, 2 Tests 15% each, Final Exam 15%, and Class Participation 10%. Because of the nature of this course, its assignments, and its assessment, your attendance and participation is both crucial and required. If you must miss a class for some reason beyond your control, talk to me about it in advance. More than two unexcused absences will lower your grade.

Outline of Weekly Readings

This outline suggests covering two or three short sections each day. This will require coverage of some material outside of class. We will modfy this outline as we go along.

Week 1: The algebra and geometry of complex numbers, polar and exponential forms. Sections 1--8.
Week 2: Roots, topology, functions of a complex variable, and limits. Sections 9--15.
Week 3: Limits, continuity, derivatives, and the Cauchy-Riemann equations. Sections 16--21.
Week 4: Differentiability, polar form, analytic and harmonic functions, the reflection principle. Sections 22--28.
Week 5: Elementary functions (exponential, log, trig, and hyperbolic). Sections 29--36.
First Test about now.
Week 6: Preparation for contour integrals. Sections 37--40.
Week 7: Contour integrals, antiderivatives. Sections 41--44.
Week 8: The Cauchy-Goursat Theorem and extensions. Sections 45--49.
Week 9: The Cauchy integral formula and its consequences. The fundamental theorem of algebra, and the maximum modulus principle. Sections 50--54.
Week 10: Liouville's Theorem and the Fundamental Theorem of Algebra. Sections 39--42.
Week 11: Sequences and series. Taylor series, Laurent series. Sections 55--61.
Second Test about now.
Week 12: Power series, absolute and uniform convergence, integration and differentiation of series. Sections 62--66.
Week 13: Residues and the residue theorem, poles. Sections 68--74.
Week 14: Zeroes and poles, evaluating real improper integrals. Sections 75--79.
Final Exam: Tuesday, December 13, 2011 1:30 PM.

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