Complex Analysis - Maths KU

C
C
y
Figure 5.34 Figure for Problem 9
C
y
2
x 4 + y 4 = 16
Figure 5.35 Figure for Problem 10
y
Figure 5.36 Figure for Problem 23
1
x
x
x
5.3 Cauchy-Goursat Theorem 263
sin z
5. f(z) =
(z2 − 25)(z2 e
6. f(z) =
+9)
z
2z2 +11z +15
7. f(z) = tan z 8. f(z) = z2 − 9
cosh z
�
1
9. Evaluate dz, where C is the contour shown in Figure 5.34.
C z
�
5
10. Evaluate
dz, where C is the contour shown in Figure 5.35.
z +1+i
C
In Problems 11–22, use any of the results in this section to evaluate the given integral
along
�
the
�
indicated closed contour(s).
11. z +
C
1
�
� �
dz; |z| =2 12. z +
z
C
1
z2 �
dz; |z| =2
�
z
13.
C z2 dz; |z| =3
− π2 �
10
14.
dz; |z + i| =1
C (z + i) 4
�
2z +1
15.
C z2 1
dz; (a) |z| = , (b) |z| =2, (c) |z − 3i| =1
2 + z
�
2z
16.
C z2 dz; (a) |z| =1, (b) |z − 2i| =1, (c) |z| =4
+3
�
−3z +2
17.
C z2 dz; (a) |z − 5| =2, (b) |z| =9
− 8z +12
� � �
3 1
18.
− dz; (a) |z| =5, (b) |z − 2i| =
C z +2 z − 2i
1
2
�
z − 1
1
19.
dz; |z − i| = 2
C z(z − i)(z − 3i)
�
1
20.
C z3 dz; |z| =1
+2iz2 �
21. Ln(z + 10) dz; |z| =2
C
� �
�
5 3 10
22.
+ − + 7 csc z dz; |z − 2| =
C (z − 2) 3 (z − 2) 2 z − 2 1
2
�
8z − 3
23. Evaluate
C z2 dz, where C is the “figure-eight” contour shown in Figure
− z
5.36. [Hint: Express C as the union of two closed curves C1 and C2.]
24. Suppose z0 is any constant complex number interior to any simple closed curve
contour C. Show that for a positive integer n,
⎧
�
⎨
dz 2πi, n =1
=
C (z − z0) n ⎩ 0, n > 1.
In Problems 25 and 26, evaluate the given contour integral by any means.
25.
� � �
z
e
− 3¯z dz, where C is the unit circle |z| =1
z +3
C