This preview shows page 1. Sign up to view the full content.
Unformatted text preview: f (z ) dz + f (z ) dz =
Γ Γ2 Γ1 f (z ) dz = 0 f (z ) dz − = −Γ2 Γ1 Section 4.4
9. Which of the following domains are simply connected?
(a) The horizontal strip  Im z  < 1
Solution: This is simply connected.
(b) The annulus 1 < z  < 2
Solution: This is not simply connected.
(c) the set of all point in the plane except those on the nonpositive xaxis.
Solution: This is simply connected.
12. Given that D is a domain containing the closed contour Γ, that z0 is a point not in
(z − z0 )−1 dz = 0, explain why D is not simply connected. D , and that
Γ Solution: If D were simply connected the result would contradict Cauchy’s Integral
Theorem. 3 13. Evaluate
below. 1
dz along the three closed contours Γ1 , Γ2 , and Γ3 in the ﬁgure
z2 + 1 Solution: We have 1
z 2 +1 Γ1 Γ2 Γ3 dz = i/2
z +i − i/2
.
z −i Thus, i
i
1
1
1
dz =
dz −
dz
+1
2 Γ1 z + i
2 Γ1 z − i
i
i
= (0) − (2πi) = π
2
2
1
i
i
1
1
dz =
dz −
dz
2+1
z
2 Γ2 z + i
2 Γ2 z − i
i
i
= (2πi) − (2πi) = 0
2
2
1
1
i
i
1
dz =
dz −
dz
2+1
z
2 Γ3 z + i
2 Γ3 z − i
i
i
= (2πi) − (0) = −π
2
2
z2 16. Show that if f is of the form
Ak Ak−1
A1
+ k −1 + · · · +
+ g (z ) (k ≥ 1)
zk
z
z
where g is analytic inside and on the circle z  = 1, then
f (z ) = f (z ) dz = 2πiA1
z =1 Solution: We have
f (z ) dz =
z =1 Ak
dz + · · · +
k
z =1 z A1
dz +
z =1 z g (z ) dz
z =1 = 0 + · · · + 0 + 2πiA1 + 0
The ﬁrst k − 1 integrals are 0 by path independence, the last integral is zero by Cauchy’s
Integral Theorem, and the remaining integral is 2πi by the theorem from class.
4...
View
Full
Document
This note was uploaded on 09/17/2013 for the course AMATH 332 taught by Professor R.moraru during the Spring '11 term at Waterloo.
 Spring '11
 R.Moraru

Click to edit the document details