Chapter 14 - Complex Variables

Abstract

Extending functional analysis into the complex plane gives a more elegant and complete understanding of functions, even those involving only real variables. Contour integrals in the complex plane obey some very remarkable relations, including Cauchy’s theorem and Cauchy’s integral formula. Many real integrals can be most easily evaluated by transforming them into contour integrals. The complex forms of Taylor and Laurent series provide a deeper understanding of their properties. All special functions have contour-integral representations, which can be useful in deriving relations between them.