Complex Analysis - Maths KU

272 Chapter 5 Integration in the Complex Plane
� 2πi
� 1+(π/2)i
15. cosh zdz 16.
sinh 3zdz
17.
18.
19.
20.
�
�
πi
C
C
� 4i
1
z dz, C is the arc of the circle z =4eit , −π/2 ≤ t ≤ π/2
1
dz, C is the line segment between 1 + i and 4 + 4i
z
−4i
� √
1+ 3i
1−i
1
dz, C is any contour not passing through the origin
z2 �
1 1
+
z z2 �
dz, C is any contour in the right half-plane Re(z) > 0
In Problems 21–24, use integration by parts (13) to evaluate the given integral.
Write each answer in the form a + ib.
21.
� i
e
π
z cos zdz 22.
� i
z sin zdz
23.
� 1+i
ze
0
z dz 24.
� πi
z 2 e z dz
i
In Problems 25 and 26, use Theorem 5.7 to evaluate the given integral. In each
integral z 1/2 is the principal branch of the square root function. Write each answer
in the form a + ib.
�
1
25.
C 4z1/2 dz, C is the arc of the circle z =4eit , −π/2 ≤ t ≤ π/2
�
26. 3z 1/2 dz, C is the line segment between z0 = 1 and z1 =9i
C
Focus on Concepts
27. Find an antiderivative of f(z) = sin z 2 . Do not think profound thoughts.
28. Give a domain D over which f(z) =z(z +1) 1/2 is analytic. Then find an
antiderivative of f in D.
5.5 Cauchy’s Integral Formulas and
Their Consequences
In the last two5.5 sections we saw the importance of the Cauchy-Goursat theorem in the
evaluation of contour integrals. In this section we are going to examine several more consequences
of the Cauchy-Goursat theorem. Unquestionably, the most significant of these is
the following result:
The value of a analytic function f at any point z0 in a simply connected domain
can be represented by a contour integral.
i
0