DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KANSAS
MIDTERM MATH 800 - Spring 2012
Your Name:
1
(60)
2
(100)
3
(120)
4
(120)
Total (400)
2
(1) (60 points)
Suppose is an open set and P0 D(P0 , r) . Let f : \ cfw_P0 C is
holomorphic, so that for soem constant C
SOLUTION OF SOME PROBLEMS FROM HOMEWORK SET I
Problem 46/page 26
By assumption u C 2 , hence h C 1 . Also, there is the equality uxy = uyx . Now,
we need to check Cauchy-Riemann. We have
( h)x = uyx = uxy = ( h)y
( h)y = uyy = uxx = ( h)x .
Problem 53/pag
SOLUTION OF SOME PROBLEMS FROM HOMEWORK SET II
Problem 16/page 62
Letting h = f g : U \ cfw_0 C be a holomorphic function, so that h (z ) = 0. For
any two points a, b U \ cfw_0, there is a smooth curve that connects them and does
not go through the origin
SOLUTION OF SOME PROBLEMS FROM HOMEWORK SET III
Problem 10, page 95
Starting with
1
=
1w
take two derivatives. We obtain
2
=
(1 w)3
wk ,
k=0
k (k 1)wk2 ,
k=2
2
Taking w = z and multiplying the result by z 2 yields
z2
=
(1 z 2 )3
k=2
k (k 1) 2k2
z
2
The ra
SOLUTION OF SOME PROBLEMS FROM HOMEWORK SET IV
1/page 145
Denote the zeroes of the polynomial p(z ) by Q1 , . . . , Qm . Consider the function
f (z )
,
R (z )
g (z ) =
in C \cfw_P1 , . . . , Pk , Q1 , . . . , Qm Note that f is meromorpic, with all singul
SELECTED PROBLEMS - SOLUTIONS
Problem 17/page 204 Denote 1 := , 2 = , . . . , j +1 := j . These
are all maps from into itself. We will show rst that j (P ) = 0 for all
j 1. We have
j +1 (z ) = j (z )( (z )2 + j (z ) (z ).
For z = P , taking into account