Page 113 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 10.5 Self Assessment

1. A ................... z of f is said to be isolated if there is a neighborhood of z which contains no
0
0
singular points of f save z .
0
2. ................... says that the integral of f is simply 2i times the sum of the residues at the
singular points enclosed by the contour C.
3. In order for the ................... to be of much help in evaluating integrals, there needs to be
some better way of computing the residue—finding the Laurent expansion about each
isolated singular point is a chore.
4. Suppose z is an isolated singularity of f and suppose that the ................... of f at z contains
0
0
only a finite number of terms involving negative powers of z – z . Thus,
0
c c c


f(z)   n   n 1  ...   1  c  c (z z ) ...

(z z ) n (z z ) n 1 (z z ) 0 1 0




0
0
0
10.6 Review Questions
1. Evaluate the integrals. In each case, C is the positively oriented circle |z| = 2.
(a)  e 1/z 2 dz.
C
1 
(b)  sin    dz.
C  z 
1 
(c)  cos    dz.
C  z 
1  1 
(d)  sin   dz.
C z  z 
1 
(e)  1 cos    dz.
C z  z 
2. Suppose f has an isolated singularity at z . Then, of course, the derivative f’ also has an
0
isolated singularity at z . Find the residue Res f'.
0
z 0 z
3. Given an example of a function f with a simple pole at z such that Res f  0, or explain
0
z 0 z
carefully, why there is no such function.
4. Given an example of a function f with a pole of order 2 at z such that Res f  0, explain
0
z 0 z
carefully, why there is no such function.
5. Suppose g is analytic and has a zero of order n at z0 (That is, g(z) = (z – z )nh(z), where
0
h(z )  0.). Show that the function f given by
0
1
f(z) 
g(z)
has a pole of order n at z . What is Res f?
0
z 0 z

106 LOVELY PROFESSIONAL UNIVERSITY