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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern