Page 46 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 4: Integration

4. Let C be the part of the circle (t) = e in the first quadrant from a = 1 to b = i. Find as small Notes
it
4
2
an upper bound as you can for (z  z  5)dz .

C
5. Evaluate f(z)dz where f(z) = z  2z and C is the path from z = 0 to z = 1 + 2i consisting of

C
the line segment from 0 to 1 together with the segment from 1 to 1 + 2i.
6. Suppose C is any curve from 0 to  + 2i. Evaluate the integral

z 
 cos    dz.
C  2 
3 5 1
7. (a) Let F(z) = log z,    argz  .  Show that the derivative F’(z) = .
4 4 z

1
(b) Let G(z) = logz,   argz  7 . Show that the derivative G’(z) = .
4 4 z
(c) Let C be a curve in the right-half plane D = {z : Rez  0} from –i to i that does not pass
1
1
through the origin. Find the integral
1
 dz.
C 1 z
(d) Let C be a curve in the left-half plane D = {z : Rez  0} from –i to i that does not pass
2
2
through the origin. Find the integral.
1
 dz.
C 2 z
8. Let C be the circle of radius 1 centered at 0 with the clockwise orientation. Find

1
 z dz.
C
9. (a) et H(z) = z , – < arg z < . Find the derivative H’(z).
c

(b) Let K(z) = z ,    argz  7 . Find the derivative K’(z).
c
4 4
(c) Let C be any path from –1 to 1 that lies completely in the upper half-plane and does
not pass through the origin. (Upper half-plane {z : Imz  0}.) Find

 F(z)dz,
C
where F(z) = z , – < arg z  .
i
10. Suppose P is a polynomial and C is a closed curve. Explain how you know that P(z)dz  0.

C

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