c - Chapter 16 Complex Analysis The term"complex analysis...

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Chapter 16 Complex Analysis The term “complex analysis” refers to the calculus of complex-valued functions f ( z ) depending on a single complex variable z . On the surface, it may seem that this subject should merely be a simple reworking of standard real variable theory that you learned in first year calculus. However, this na¨ ıve first impression could not be further from the truth! Complex analysis is the culmination of a deep and far-ranging study of the fun- damental notions of complex differentiation and complex integration, and has an elegance and beauty not found in the more familiar real arena. For instance, complex functions are always analytic , meaning that they can be represented as convergent power series. As an immediate consequence, a complex function automatically has an infinite number of derivatives, and difficulties with degree of smoothness, strange discontinuities, delta func- tions, and other forms of pathological behavior of real functions never arise in the complex realm. The driving force behind many applications of complex analysis is the remarkable and profound connection between harmonic functions (solutions of the Laplace equation) of two variables and complex functions. Namely, the real and imaginary parts of a complex analytic function are automatically harmonic. In this manner, complex functions provide a rich lode of new solutions to the two-dimensional Laplace equation to help solve boundary value problems. One of the most useful practical consequences arises from the elementary observation that the composition of two complex functions is also a complex function. We interpret this operation as a complex changes of variables, also known as a conformal map- ping since it preserves angles. Conformal mappings can be effectively used for constructing solutions to the Laplace equation on complicated planar domains, and play a particularly important role in the solution of physical problems. Complex integration also enjoys many remarkable properties not found in its real sibling. Integrals of complex functions are similar to the line integrals of planar multi- variable calculus. The remarkable theorem due to Cauchy implies that complex integrals are generally path-independent — provided one pays proper attention to the complex singularities of the integrand. In particular, an integral of a complex function around a closed curve can be directly evaluated through the “calculus of residues”, which effectively bypasses the Fundamental Theorem of Calculus. Surprisingly, the method of residues can even be applied to evaluate certain types of definite real integrals. In this chapter, we shall introduce the basic techniques and theorems in complex analysis, paying particular attention to those aspects which are required to solve boundary value problems associated with the planar Laplace and Poisson equations. Complex anal- ysis is an essential tool in a surprisingly broad range of applications, including fluid flow, 2/25/07 872 c circlecopyrt 2006 Peter J. Olver
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