Complex Analysis - Maths KU

256 Chapter 5 Integration in the Complex Plane
27.
28.
�
�
C
C
(z 2 +4)dz, where C is the line segment from z =0toz =1+i
1
dz, where C is one-quarter of the circle |z| = 4 from z =4i to z =4
z3 Focus on Concepts
29. (a) Use Definition 5.3 to show for any smooth curve C between z0 and zn �
that
dz = zn − z0.
C
(b) Use the result in part (a) to verify your answer to Problem 14.
(c) What is �
dz when C is a simple closed curve?
C
30. Use Definition 5.3 to show for any smooth curve C between z0 and zn �
that
1
zdz = C 2 (z2 n − z 2 0). [Hint: The integral exists. So choose z ∗ k = zk and
z ∗ k = zk–1.]
31. Use the results of Problems 29 and 30 to evaluate �
(6z +4)dz where C is:
(a) The straight line from 1 + i to2+3i.
(b) The closed contour x 4 + y 4 =4.
�
1
32. Find an upper bound for the absolute value of the integral
C z2 dz, where
+1
�the
contour C is the line segment from z =3toz =3+i. Use the fact that
�z 2 +1 � � = |z − i||z + i| where |z − i| and |z + i| represent, respectively, the
distances from i and −i to points z on C.
33. Find an upper bound for the absolute value of the integral �
Ln(z +3)dz,
C
where the contour C is the line segment from z =3i to z =4+3i.
5.3 Cauchy-Goursat Theorem
In this section 5.3 we shall concentrate on contour integrals, where the contour C is a simple
closed curve with a positive (counterclockwise) orientation. Specifically, we shall see that
when f is analytic in a special kind of domain D, the value of the contour integral �
C
C
f(z) dz
is the same for any simple closed curve C that lies entirely within D. This theorem, called
the Cauchy-Goursat theorem, is one of the fundamental results in complex analysis.
Preliminary to discussing the Cauchy-Goursat theorem and some of its ramifications,
we need to distinguish two kinds of domains in the complex plain: simply connected and
multiply connected.
Simply and Multiply Connected Domains Recall fromSection
1.5 that a domain is an open connected set in the complex plane. We say
that a domain D is simply connected if every simple closed contour C lying
entirely in D can be shrunk to a point without leaving D. See Figure 5.26.
In other words, if we draw any simple closed contour C so that it lies entirely
within a simply connected domain, then C encloses only points of the domain