Page 52 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 5: Cauchy’s Theorem

5.6 Review Questions Notes

1. Suppose C is homotopic to C in D, and C is homotopic to C in D. Prove that C is
1
1
2
3
2
homotopic to C in D.
3
2. Explain how you know that any two closed curves in the plane C are homotopic in C.
3. A region D is said to be simply connected if every closed curve in D is contractible to a
point in D. Prove that any two closed curves in a simply connected region are homotopic
in D.
4. Prove Cauchy’s Theorem.
5. Let S be the square with sides x = ± 100, and y = ± 100 with the counterclockwise orientation.
1
Find  dz.
s z

6. (a) Find  1 dz, where C is any circle centered at z = 1 with the usual counterclockwise

C z 1
orientation: (t) = 1 + Ae , 0  t  1.
2it
1
(b) Find  dz, where C is any circle centered at z = –1 with the usual

C z 1
counterclockwise orientation.
1
(c) Find 2  dz, where C is the ellipse 4x + y = 100 with the counterclockwise
2
2
C z – 1
orientation. [Hint: partial fractions]
1
(d) Find 2  dz, where C is the circle x – 10x + y = 0 with the counterclockwise
2
2
C z – 1
orientation.
7. Evaluate Log(z 3)dz, where C is the circle |z| = 2 oriented counterclockwise.

C

8. Evaluate 1 n  dz where C is the circle described by (t) = e 2it , 0  t  1, and n is an
C z
integer  1.
1
9. (a) Does the function f(z) = have an antiderivative on the set of all z  0? Explain.
z

1
(b) How about f(z) = , n an integer  1?
z n

2
z
10. Find as large a set D as you can so that the function e have an antiderivative on D.
11. Explain how you know that every function analytic in a simply connected region D is the
derivative of a function analytic in D.

LOVELY PROFESSIONAL UNIVERSITY 45