Page 719 - Advanced_Engineering_Mathematics o'neil

20.1 The Integral of a Complex Function 699

Finally,
9 9 27 9

Im(z)dz =− π + i + 5 + i = 5 − π + 18i.
4 2 2 4
γ
We will list some properties of the complex integral. These reflect properties of line integrals
from Chapter 12.

1. ( f (z) + g(z))dz = f (z)dz + g(z)dz.
γ γ γ

2. If c is a number, then cf (z)dz = c f (z)dz.
γ γ
3. Reversing the orientation on the curve changes the sign of the integral. Specifically, given
γ defined on [a,b], define ϕ(t) = γ(a + b − t) for a ≤ t ≤ b. Then
γ(a) = ϕ(b) and γ(b) = ϕ(a).

The initial point of γ is the terminal point of ϕ, and the terminal point of γ is the initial
point of ϕ. We denote the curve ϕ formed in this way as −γ . Then

f (z)dz =− f (z)dz.
−γ γ
4. There is a version of the fundamental theorem of calculus for complex integrals. Suppose

f is continuous on an open set G and F is defined on G with the property that F (z) =
f (z).If γ is a smooth curve in G, defined on an interval [a,b], then

f (z)dz = F(γ (b)) − F(γ (a)).
γ
To see why this is true, write F(z) = U(x, y) + iV (x, y) to get

b

f (z)dz = f (z(t))z (t)dt
γ a
b b d

= F (z(t))z (t)dt = F(z(t))dt
a a dt
d d
b b
= U(x(t), y(t))dt + i V (x(t), y(t))dt
a dt a dt
= U(x(b), y(b)) + iV (x(b), y(b)) − iU(x(a), y(a)) − iV (x(a), y(a))
= F(γ (b)) − F(γ (a)).

One ramification of this is that, under the given conditions, the value of f (z)dz
γ
depends only on the initial and terminal points of γ and not on γ itself. This is called
independence of path, and we saw a version of it in Chapter 12 with conservative force

fields. If γ is a closed curve, the initial and terminal points are the same, and f (z)dz =
γ
0. We will consider this in more detail in the next section.
EXAMPLE 20.8
2 2 3
Evaluate z dz, with γ any smooth curve from i to 1 − i. Since F (z) = z ,if F(z) = z /3,

γ
(1 − i) i −2 − i
3 3
2
z dz = F(1 − i) − F(i) = − = .
3 3 3
γ
regardless of how γ moves about the plane between i and 1 − i.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:32 THM/NEIL Page-699 27410_20_ch20_p695-714