M519: Complex Variables I  --  Spring 2006
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Course Description:

Complex variables is a beautiful area from a purely mathematical point of view, as well as a powerful tool for solving a wide array of applied problems. It is related to many mathematical disciplines, including in particular real analysis, differential equations, algebra and topology. The numerous applications include all kinds of wave propagation phenomena such as those occurring in electrodynamics, optics, fluid mechanics and quantum mechanics, diffusion problems such as heat and contaminant diffusion, engineering tasks such as the computation of buoyancy and resistance of wings, the flows in turbines and the design of optimal car bodies, and signal processing and communication theory.

 

Historically, complex numbers originated from the desire to find a uniform representation of solutions of algebraic equations. From this perspective, the field of complex numbers is a natural extension of the field of real numbers with the property that it is algebraically closed, that is, every polynomial can be factorized into linear polynomials. After analysis has been introduced, it became a natural task to extend the concepts of differential and integral calculus and function series to complex variables. In the 19th century complex analysis emerged as an independent mathematical discipline, most notably through the work of Augustin-Louis Cauchy (1789-1857), Karl Weierstrass (1815-1897), and Bernhard Riemann (1826-1866).

The purpose of this course is an introduction to the theory and application of complex variables and complex functions. Part I is devoted to the basic mathematical theory, and Part II to selected applications.

          Part I: Fundamentals and Techniques of Complex Function Theory

We begin, in Chapter 1, by introducing complex numbers, elementary complex functions, and concepts from analysis such as limits, continuity and differentiability. It will be seen that complex numbers have a simple two-dimensional character that admits a straightforward geometric description. While many results of real analysis carry over, some important novel notions appear in the calculus of complex functions. Applications to differential equations are briefly discussed.

In Chapter 2, we study the notion of analytic functions and their properties. It will be shown that a complex function is differentiable if and only if an important compatibility relationship between its real and imaginary parts is satisfied, which is referred to as Cauchy Riemann equations. The concepts of multivalued functions and complex integration are considered in some detail. The technique of integration in the complex plane is discussed and two very important results of complex analysis are derived: Cauchy’s theorem and a corollary: Cauchy’s integral formula.

Chapter 3 deals with sequences, series and singularities of complex functions. It turns out that the representation of complex functions frequently requires the use of infinite series expansions. The best known are the Taylor and Laurent series, which represent analytic functions in appropriate domains. Applications often require that we manipulate series by termwise differentiation and integration. These operations may be substantiated by employing the notion of uniform convergence. Series expansion breaks down at points or curves where the represented function is not analytic. Such locations are termed singular points or singularities of the function. The study of singularities is vitally important in many applications including contour integration, differential equations, and conformal mappings.

In Chapter 4 we extend Cauchy’s theorem to cases where the integrand is not analytic, for example, if the integrand has isolated singular points. Each isolated singular point contributes to what is called the residue of the singularity. This extension, called residue theorem, is very useful in applications such as the evaluation of definite integrals of various types. The residue theorem provides a straightforward and sometimes the only method to compute these integrals, which include real integrals that cannot be computed on the basis of real integral calculus alone. We also show how to use contour integrals to compute the solutions of certain partial differential equations by the techniques of Fourier and Laplace transforms.

          Part II: Applications of Complex Function Theory

A number of problems arising in fluid mechanics, electrostatics, heat conduction, and many other physical situations can be formulated in terms of Laplace’s equation in a two-dimensional domain with given boundary conditions. Conformal mappings are transformations through which a given domain is transformed to a simple domain such as a half plane, in which the problem is greatly simplified. Chapter 5 is devoted to general properties of conformal mappings and a number of applications including problems from fluid flow, steady state heat conduction, and electrostatics.

The second application concerns the asymptotic evaluation of integrals. This is motivated by the fact that the solution of a large class of physically important problems can be represented in terms of definite integrals. Although such integrals provide exact solutions, their content is often not obvious. In order to decipher their main mathematical and physical properties, it is often useful to study their behavior in the limit of large values of a certain parameter which, for example, gives insight into the far field of a scattered wave. In Chapter 6 techniques for evaluating definite integrals asymptotically are discussed and related to other methods such as the WKB method. The most well-known methods to study integrals containing a large parameter are Laplace’s method, the method of stationary phase, and the steepest descent method.