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**Unformatted text preview: **Chapter 7 Complex Analysis and Conformal Mapping The term “complex analysis” refers to the calculus of complex-valued functions f ( z ) depending on a single complex variable z . To the novice, 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 funda- mental notions of complex differentiation and integration, and has an elegance and beauty not found in the real domain. For instance, complex functions are necessarily analytic , meaning that they can be represented by convergent power series, and hence are infinitely differentiable. Thus, difficulties with degree of smoothness, strange discontinuities, subtle convergence phenomena, and other pathological properties of real functions never arise in the complex regime. The driving force behind many of the applications of complex analysis is the re- markable connection between complex functions and harmonic functions of two variables, a.k.a. solutions of the planar Laplace equation. To wit, the real and imaginary parts of any complex analytic function are automatically harmonic. In this manner, complex functions provide a rich lode of additional solutions to the two-dimensional Laplace equation, which can be exploited in a wide range of physical and mathematical applications. One of the most useful consequences stems from the elementary observation that the composition of two complex functions is also a complex function. We re-interpret this operation as a complex change of variables, producing a conformal mapping that preserves angles. Con- formal mappings can be effectively used for constructing solutions to the Laplace equation...

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