chapter 7, Complex Integration, Laurent Series, and Residues

Excerpt

In this chapter we extend the discussion of complex variables to complex integrals and Laurent series. Complex integrals in general behave similar to the line integrals found in vector analysis. However, the method of residue calculus used for evaluating integrals that involve an analytic function is more powerful than any counterpart in vector analysis—i.e., the residue calculus can be used to evaluate inverse Laplace transforms, and is also useful in calculating Fourier and Mellin transform integrals. Laurent series are a natural generalization of Taylor series that provide the basis for developing the residue calculus.

7.1 Introduction

The two-dimensional nature of a complex variable required us in Chapter 6 to generalize our notion of derivative in the complex plane. This was a consequence of the fact that a complex variable can approach its limit value from infinitely-many directions, rather than just two directions as in the case of a real variable. This two-dimensional aspect of a complex variable will also influence the theory of integration in the complex plane, requiring us to consider integrals along general curves in the plane rather than simply along segments of the x-axis. Because of this, we find that complex integrals behave more like line integrals from vector analysis instead of like standard Riemann integrals.

The most practical application of the complex integral is in the evaluation of certain real integrals, including those that commonly appear in the use of integral transforms. A secondary reason why the complex integral is important is that the method of complex integration (also called contour integration) yields simple proofs of some basic properties of analytic functions that would otherwise be very difficult to prove. In particular, we can use complex integration to prove that an analytic function has higher-order derivatives—in fact, derivatives of all orders. Thus far, however, we have only shown that an analytic function has a first derivative.

Most of the theory of integration in the complex plane involving analytic functions is due to Augustin L. Cauchy (1789–1857), the famous French mathematician. His main result, now known as Cauchy's integral theorem, or simply Cauchy's theorem, states that the integral of an analytic function around a simple closed curve is zero. Although sounding almost trivial, most of the theory of complex integration is a consequence of this very important result.

7.2 Line Integrals in the Complex Plane

Once again, we find it necessary to introduce some special terminology that arises in the discussion of integration in the complex plane. However, there are variations of these terms that appear in some textbooks.

Let us imagine that the real and imaginary parts of the complex variable z = x+iy depend upon a real parameter t, i.e., x = x(t) and y = y(t). In this case we can write