**for**

**Chapter 6 Complex
Integration**

**Overview**

Of the two main topics studied in calculus -
differentiation and integration - we have so far only studied
derivatives of complex functions. We now turn to the problem of
integrating complex functions. The theory you will learn is elegant,
powerful, and a useful tool for physicists and engineers. It also
connects widely with other branches of mathematics. For example, even
though the ideas presented here belong to the general area of
mathematics known as analysis, you will see as an application of them
one of the simplest proofs of the fundamental theorem of algebra.

**6.1 Complex
Integrals**

In Section 3.1 we saw how the derivative of a complex function is defined. 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

**Definition 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

(6-1) .

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 and , we
have

(6-2) .

**Example 6.1.** Show
that .

Solution. We write the integrand in terms of its real
and imaginary parts, i.e., . Here, and . The
integrals of u(t) and v(t)
are

, and

.

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 . Likewise, . Therefore,

.

Complex integrals have properties that are similar to those of real integrals. We now trace through several commonalities. Let and be continuous on .

Using Definition (6-1), we can easily
show that the integral of their sum is the sum of their integrals,
that is

(6-3) .

If we divide the interval into and and
integrate f(t) over these
subintervals by using (6-1), then we
get

(6-4) .

Similarly, if denotes
a complex constant, then

(6-5) .

If the limits of integration are reversed, then

(6-6) .

The integral of the product f(t)g(t)
becomes

(6-7)

**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

Therefore, .

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 . Since
, equation
(6-2) takes on the familiar form

(6-8) .

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):

(6-9) .

**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.

**Extra Example
1.** Show that .

**Explore
Solution for Extra Example 1.**

**Exercises
for Section 6.1. Complex
Integrals**** **

**The Next Module
is**

**Contours
and Contour Integrals**

**Return to the Complex
Analysis Modules**

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell