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 näıve first impression could not be further from the truth! Complex analysis is the culmination of a deep and far-ranging study of the fundamental 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 realm. The driving force behind many of the applications of complex analysis is the remarkable 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 (signed) angles in the Euclidean plane. Conformal mappings can be effectively used for constructing solutions to the Laplace equation on complicated planar domains that appear in a wide range of physical problems, including fluid mechanics, aerodynamics, thermomechanics, electrostatics, and elasticity. In this chapter, we will develop the basic techniques and theorems of complex analysis that impinge on the solution to boundary value problems associated with the planar Laplace and Poisson equations. We assume the reader is familiar with the basics of complex numbers and complex arithmetic (as in [16; Appendix A]), and commence our exposition with the basics of complex functions and their differential calculus. We then proceed to develop the theory and applications of conformal mappings. The final section contains a brief introduction to complex integration and a few of its applications. Further developments and additional details and results can be found in a wide variety of texts devoted to complex analysis, including [1, 10, 18, 19].