Page 732 - Advanced_Engineering_Mathematics o'neil

712 CHAPTER 20 Complex Integration

y

4i
2

x
–2
1

FIGURE 20.12 , γ 1 , and γ 2 in Exam-
ple 20.15.

z z z

dz = dz + dz.
(z + 2)(z − 4i) γ 1 (z + 2)(z − 4i) γ 2 (z + 2)(z − 4i)
Use a partial fractions decomposition to write
z 1 − 2i 1 4 + 2i 1
= + .
(z + 2)(z − 4i) 5 z + 2 5 z − 4i
Putting the last two equations together with Cauchy’s theorem and the conclusion of Exam-
ple 20.11, we have

z 1 − 2i 1 4 + 2i 1
dz = dz + dz
(z + 2)(z − 4i) 5 γ 1 z + 2 γ 1 5 z − 4i
1 − 2i 1 4 + 2i 1
+ dz + dz
5 z + 2 5 z − 4i
γ 2 γ 2
1 − 2i 4 + 2i
= (2πi) + (2πi)
5 5
=2πi.

A proof of the extended deformation theorem can be modeled after that of the deformation
theorem, except now we draw a line segment from to γ 1 , from γ 1 to γ 2 and so on until we come
to γ n−1 to γ n , and finally from γ n to , as in Figure 20.13 for n = 3.

y
L 4

3
1 L 3
L 1

L 2 2

x

FIGURE 20.13 Argument for the extended deforma-
tion theorem.

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October 14, 2010 15:32 THM/NEIL Page-712 27410_20_ch20_p695-714