Complex Analysis - Maths KU

z 0
C 1
D
z 1
C
Figure 5.38 If f is analytic in D,
integrals on C and C1 are equal.
5.4 Independence of Path 265
Note that � b
f(x) dx depends only on the numbers a and b at the initial and terminal points
a
of the interval of integration. In contrast, the value of a real line integral �
Pdx+ Qdy
C
generally depends on the curve C. However, there exist integrals �
Pdx+ Qdy whose
C
value depends only on the initial point A and terminal point B of the curve C, and not
on C itself. In this case we say that the line integral is independent of the path. For
example, �
ydx+ xdy is independent of the path. See Problems 19–22 in Exercises 5.1.
C
A line integral that is independent of the path can be evaluated in a manner similar to (1).
It seems natural then to ask:
Is there a complex version of the fundamental theorem of calculus?
Can a contour integral �
f(z) dz be independent of the path?
C
In this section we will see that the answer to both of these questions is yes.
Path Independence The definition of path independence for a
contour integral �
�
f(z) dz is essentially the same as for a real line integral
C
Pdx+ Qdy.
C
Definition 5.4 Independence of the Path
Let z0 and z1 be points in a domain D. A contour integral �
f(z) dz
C
is said to be independent of the path if its value is the same for all
contours C in D with initial point z0 and terminal point z1.
At the end of the preceding section we noted that the Cauchy-Goursat
theoremalso holds for closed contours, not just simple closed contours, in a
simply connected domain D. Now suppose, as shown in Figure 5.38, that C
and C1 are two contours lying entirely in a simply connected domain D and
both with initial point z0 and terminal point z1. Note that C joined with the
opposite curve −C1 forms a closed contour. Thus, if f is analytic in D, it
follows fromthe Cauchy-Goursat theoremthat
�
�
f(z) dz + f(z) dz =0. (2)
But (2) is equivalent to
C
�
C
−C1
�
f(z) dz =
C1
f(z) dz. (3)
The result in (3) is also an example of the principle of deformation of contours
introduced in (5) of Section 5.3. We summarize the last result as a theorem.
Theorem 5.6 Analyticity Implies Path Independence
Suppose that a function f is analytic in a simply connected domain
D and C is any contour in D. Then �
f(z) dz is independent of the
C
path C.