Complex Analysis - Maths KU

298 Chapter 5 Integration in the Complex Plane
19. If p(z) is a polynomial in z then the function f(z) =1/p(z) can be never be an
entire function.
20. The function f(z) = cos z is entire and not a constant and so must be unbounded.
In Problems 21–40, try to fill in the blanks without referring back to the text.
21. z(t) =e it2
, 0 ≤ t ≤ √ 2π, is a parametrization for a .
22. z(t) =z0 + e it , 0 ≤ t ≤ 2π, is a parametrization for a .
23. The difference between z1(t) =e it , 0 ≤ t ≤ 2π and z2(t) =e i(2π−t) , 0 ≤ t ≤ 2π
is
�
.
24. (2y + x − 6ix 2 ) dz = , where C is the triangle with vertices 0, i,
C
1+i, traversed counterclockwise.
25. �If
f is a polynomial function and C is a simple closed contour, then
f(z) dz = .
26.
27.
28.
29.
30.
�
�
�
�
�
C
C
C
C
C
C
z Im(z) dz = , where C is given by z(t) =2t + t 2 i, 0 ≤ t ≤ 1.
|z| 2 dz = , where C is the line segment for 1 − i to 1 + i.
(¯z) n dz = , where n is an integer and C is z(t) =e it , 0 ≤ t ≤ 2π.
sin z
2 dz = , where C is given by z(t) =2i +4eit , 0 ≤ t ≤ π/2.
sec zdz= , where C is |z| =1.
�
1
31.
C z(z − 1)
�
32. If f(z) =
C
dz = , where C is |z − 1| = 1
2 .
ξ 2 +6ξ − 2
ξ − z
dξ, where C is |z| = 3, then f(1 + i) = .
33. If f(z) = z 3 + e z and C is a contour z = 8e it �
, 0 ≤ t ≤ 2π, then
f(z)
dz = .
C (z + πi) 3
��
�
� �
34. If | f(z) |≤2 on the circle |z| =3, then �
� f(z) dz �
� ≤ .
35. �If
n is a positive integer and C is the contour |z| = 2, then
z −n e z dz = .
C
�
cos z
36. On |z| = 1, the contour integral dz equals
C zn for n = 2, and equals for n =3.
for n = 1, equals
37.
�
z
C
n ⎧
⎨ 0,
dz =
⎩ 2πi,
|z| =1.
if n
if n
, where n is an integer and C is the circle
C