We now turn our attention to the problem of integrating complex functions. We will find that integrals of analytic functions are well behaved and that many properties from calculus carry over to the complex case.
We introduce the integral of a complex function by defining the integral of a complex-valued function of a real variable
6.1 (Definite Integral of a Complex Integrand). Let where u(t) and v(t) are real-valued functions of the real variable t for . Then
We generally evaluate integrals of this type by finding the antiderivatives of u(t) and v(t) and evaluating the definite integrals on the right side of Equation (6-1). That is, if , we have
Example 6.1. Show that .
Solution. We write the integrand in terms of its real and imaginary parts, i.e., and . The integrals of u(t) and v(t) are
Hence, by Definition (6-1),
Example 6.2. Show that .
Solution : We use the method suggested by Definitions (6-1) and (6-2).
We can evaluate each of the integrals via integration by parts. For example,
Adding to both sides of this equation and then dividing by 2 gives . Therefore,
Complex integrals have properties that are similar to those of real integrals. We now trace through several commonalities. and .
Using Definition (6-1), we can easily show that the integral of their sum is the sum of their integrals, that is
If we divide the interval and integrate f(t) over these subintervals by using (6-1), then we get
Similarly, if denotes a complex constant, then
If the limits of integration are reversed, then
The integral of the product f(t)g(t) becomes
Example 6.3. Let us verify property (6-5). We start by writing
Using Definition (6-1), we write the left side of Equation (6-5) as
which is equivalent to
It is worthwhile to point out the similarity between equation (6-2) and its counterpart in calculus. Suppose that U and V are differentiable on
and , equation (6-2) takes on the familiar form
where . We can view Equation (6-8) as an extension of the fundamental theorem of calculus. In Section 6.4 we show how to generalize this extension to analytic functions of a complex variable. For now, we simply note an important case of Equation (6-8):
Example 6.4. Use Equation (6-8) to show that .
Solution. We seek a function F with the property that . We note that satisfies this requirement, so
which is the same result we obtained in Example 6.2, but with a lot less work.
Remark 6.1 Example 6.4 illustrates the potential computational advantage we have when we lift our sights to the complex domain. Using ordinary calculus techniques to evaluate , for example, required a lengthy integration by parts procedure (Example 6.2). When we recognize this expression as the real part of , however, the solution comes quickly. This is just one of the many reasons why good physicists and engineers, in addition to mathematicians, benefit from a thorough working knowledge of complex analysis.