Complex Analysis - Maths KU

5.4 Independence of Path 271
(ii) Iff is a real function continuous on the closed interval [a, b], then
there exists a number c in the open interval (a, b) such that
� b
f(x) dx = f(c)(b − a). (14)
a
The result in (14) is known as the mean-value theorem for definite
integrals. If f is a complex function analytic in a simply connected
domain D, it is continuous at every point on a contour C in D with
initial point z0 and terminal point z1. One might expect a result
parallel to (14) for an integral � z1
f(z) dz. However, there is no such
z0
complex counterpart.
EXERCISES 5.4 Answers to selected odd-numbered problems begin on page ANS-16.
In Problems 1 and 2, evaluate the given integral, where the contour C is given in
the figure, (a) by using an alternative path of integration and (b) by using Theorem
5.7. �
�
1. (4z − 1) dz 2. e z dz
C
i
–i
y
|z| =1
Figure 5.42 Figure for Problem 1
x
C
y
0
3 + 3i
3 + i
Figure 5.43 Figure for Problem 2
In Problems 3 and 4, evaluate the given integral along the indicated contour C.
�
3. 2zdz, where C is z(t) =2t 3 + i(t 4 − 4t 3 +2), −1 ≤ t ≤ 1
4.
�
C
C
2zdz, where C is z(t) =2cos 3 πt − i sin 2 π
t, 0 ≤ t ≤ 2
4
In Problems 5-20, use Theorem 5.7 to evaluate the given integral. Write each answer
in the form a + ib.
5.
� 3+i
z
0
2 dz 6.
� 1
(3z
−2i
2 − 4z +5i) dz
7.
� 1+i
z
1−i
3 dz 8.
� 2i
(z
−3i
3 − z) dz
9.
� 1−i
(2z +1)
−i/2
2 dz 10.
� i
(iz +1)
1
3 dz
11.
� i
e
i/2
πz dz 12.
� 1+2i
ze
1−i
z2
dz
13.
� π+2i
sin z
dz
2
14.
� πi
cos zdz
π
1−2i
x