Complex Analysis - Maths KU

236 Chapter 5 Integration in the Complex Plane
x 1 *
x k *
a =x0 x1 xk–1 xk xn = b
Figure 5.1 Partition of [a, b] with x ∗ k
in each subinterval [xk−1, xk]
5.1 Real Integrals
To the Instructor: 5.1 We present this section as a review of the definitions and methods
of evaluation of the definite integral and line integrals in the plane. In our experience we
have found that a re-examination of this material contributes to a smoother introduction to
complex integration. You can, of course, skip this section and move directly into complex
contour integrals if you think your students have adequate familiarity with these concepts.
However, the terminology about curves in the plane introduced in this section will be used
in the succeeding section.
Definite Integral It is likely that you have retained at least two associations
fromyour study of elementary calculus: the derivative with slope,
and the definite integral with area. But as the derivative f ′ (x) of a real function
y = f(x) has other uses besides finding slopes of tangent lines, so too
the value of a definite integral � b
f(x) dx need not be area “under a curve.”
a
Recall, if F (x) is an antiderivative of a continuous function f, that is, F is a
function for which F ′ (x) =f(x), then the definite integral of f on the interval
[a, b] is the number
� b
a
f(x) dx = F (x)| b
a = F (b) − F (a). (1)
For example, � 2
−1 x2dx = 1
3x3�� 2
8 =
−1 3 −� − 1
�
3 =3. Bear in mind that the fun-
damental theorem of calculus, just given in (1), is a method of evaluating
� b
a f(x) dx; it is not the definition of � b
f(x) dx.
a
In the discussion that follows we present the definitions of two types of
real integrals. We begin with the five steps leading to the definition of the
definite (or Riemann) integral of a function f; we follow it with the definition
of line integrals in the Cartesian plane. Both definitions rest on the limit
concept.
Steps Leading to the Definition of the
Definite Integral
1. Let f be a function of a single variable x defined at all points in a
closed interval [a, b].
2. Let P be a partition:
a = x0