Complex Analysis. Exercise 4.
Lecturer: Prof. M. Farber. Instructor: A. Aizenbud.
1.
Semester C 2001/2002 Date: 6/2/16
For a given function f(z) and a given counter C computer the
 f ( z )dz
C
2.
a.
f ( z)
C
  e z
boundary of the square with vertices at the points 0, 1, 1+i
and i. The orientation of C be counterclockwise direction.
b.
f ( z)
C
branch | z | > 0, 0 < arg ( z ) < 2
 z 1i  e( 1i )ln z
positively oriented unit circle | z | = 1
Let C0 denote the circles |z - z0| = R taken counterclockwise. Use the parametric
representation z = z0 + R e i ( -) for C0 to derive the following integration
formula:
2Ra
a 1
(
z

z
)
dz

i
sin(a )
0

a
C0
where
3.
*
*
*
a is any real number except 0.
Principle part of the integrand is taken
Principle part of R a is taken
By finding an antiderivative calculate the following integrals, where the path is an
arbitrary contour between the indicated limits of integration:
i/2
a.

  2i
e z dz
b.
i
4.
cos( z / 2) dz
0
Let C1 denote the positively oriented circle | z | = 4 and C2 – positively oriented
boundary of the square, whose lines lie along the lines x = 1 and y = 1. prove that
 f ( z )dz  
C1
when
5.

f ( z )dz
C2
f ( z) 
1
3z  1
2
Let C be any simple closed contour, described in the positive sense in the z
plane, and
z3  2z
dz
3
(
z

w
)
C
g ( w)  
6 iw
Prove that g ( w)  
0
6.


w is inside C
otherwise
Prove that if f(z) is analytic within and on the simple closed contour C, and z0 is not
on C, then
f '( z )
 zz
C
0
f ( z)
dz
( z  z0 ) 2
C
dz  