Complex Analysis - Maths KU

C 1
y
2i
C 2
–2 2
Figure 5.58 Figure for Problem 14
x
Chapter 5 Review Quiz 297
Answers to selected odd-numbered problems begin
CHAPTER 5 REVIEW QUIZ
on page ANS-17.
In Problems 1–20, answer true or false. If the statement is false, justify your answer
by either explaining why it is false or giving a counterexample; if the statement is
true, justify your answer by either proving the statement or citing an appropriate
result in this chapter.
1. If z(t), a ≤ t ≤ b, is a parametrization of a contour C and z(a) =z(b), then C
is a simple closed contour.
2. The real line integral �
C (x2 + y 2 ) dx +2xy dy, where C is given by y = x 3 from
(0, 0) to (1,1), has the same value on the curve y = x 6 from (0, 0) to (1,1).
3. The sector defined by −π/6 < arg(z) 0 between z =1+i and z =10+8i.
�
10. If g is entire, then
�
g(z)
dz =
z − i
g(z)
dz, where C is the circle |z| = 3 and
z − i
11.
C
C1 is the ellipse x 2 + 1
9 y2 =1.
C1
�
1
dz = 0 for every simple closed contour C that encloses the
C (z − z0)(z − z1)
points z0 and z1.
12. If f is analytic within and on the simple closed contour C and z0 �
is a point
f
within C, then
C
′ �
(z)
f(z)
dz =
dz.
z − z0
C (z − z0) 2
13. �
Re (z) dz is independent of the path C between z0 = 0 and z1 =1+i.
C
14. � � � � 3 2 � � 3
4z − 2z +1 dz = 4x − 2x +1 dx, where the contour C is com-
C
−2
prised of segments C1 and C2 shown in Figure 5.58.
15. �
C1 zn dz = �
C2 zn dz for all integers n, where C1 is z(t) =e it , 0 ≤ t ≤ 2π and
C2 is z(t) =Re it , R > 1, 0 ≤ t ≤ 2π.
16. If f is continuous on the contour C, then �
f(z) dz+� f(z) dz =0.
C −C
17. On any contour C with initial point z0 = −i and terminal point z1 = i that
lies in a simply
�
connected domain D not containing the origin or the negative
i
1
real axis, dz = Ln(i)− Ln(−i) =πi.
−i z
�
1
18.
z2 +1 dz = 0, where C is the ellipse x2 + 1
4 y2 =1.
C
C