wu :: forums
« wu :: forums - the Extended Hurwitz's Theorem »

Welcome, Guest. Please Login or Register.
Jan 27th, 2022, 4:01pm

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   general
   complex analysis
(Moderators: Icarus, Eigenray, william wu, SMQ, Grimbal, towr, ThudnBlunder)
   the Extended Hurwitz's Theorem
« Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: the Extended Hurwitz's Theorem  (Read 10655 times)
immanuel78
Newbie
*





   
Email

Gender: male
Posts: 23
the Extended Hurwitz's Theorem  
« on: Oct 26th, 2008, 7:49am »
Quote Quote Modify Modify

The Hurwitz's theorem that  I have seen in the textbook until now is as follows :
 
H(G) = the set of analytic functions in G
M(G) = the set of meromorphic functions in G
C = the set of complex numbers
C infinite = C union {infinite}
 
Let {fn} be a sequence in H(G) and fn -> f , where f:G->C is continuous.
 
If f is identically not zero, closed disk B(a;R) in G and f(z) not zero in |z-a|=R ,
then there in an integer N such that for n>= N, f and fn have the same number of zeros in open disk B(a;R).
 
Now I think that {fn} in H(G) can be extended to {fn} in M(G).  
That is, Let {fn} in M(G) and fn -> f , where f : G -> C infinite is continuous
 
If f is identically not zero or infinite, closed disk B(a;R) in G  and f(z) not zero or infinite in |z-a|=R ,
then there in an integer N such that for n>= N,  
[the number of zeros of f in B(a;R)] - [the number of poles of f in B(a;R)] =
[the number of zeros of fn in B(a;R)] - [the number of poles of fn in B(a;R)].
 
Have you seen this extened theorem in your textbook or exercises?
« Last Edit: Oct 26th, 2008, 10:10pm by immanuel78 » IP Logged
Michael Dagg
Senior Riddler
****






   


Gender: male
Posts: 500
Re: the Extended Hurwitz's Theorem  
« Reply #1 on: Nov 3rd, 2008, 8:09pm »
Quote Quote Modify Modify

Start with p. 153 in Conway's book.  I believe the problem  
you mention before regarding Hardy's Theorem  came form  
Conway (actually its problem from the book?).
 
IP Logged

Regards,
Michael Dagg
immanuel78
Newbie
*





   
Email

Gender: male
Posts: 23
Re: the Extended Hurwitz's Theorem  
« Reply #2 on: Nov 29th, 2008, 5:26am »
Quote Quote Modify Modify

Theorem I mentioned above is correct, but I came to know actually a stronger result is true.  
 
There is an integer N such that for n>=N, the number of zeroes of fn in B(a;R) equals the number of zeroes of f in B(a;R), and also the number of poles of fn in B(a;R) equals the number of poles of f in B(a;R).
 
A statement and a proof of Hurwitz's theorem for meromorphic functions can be found in the book Complex Function Theory by Maurice Heins
(Academic Press, 1968 ), Theorem 4.4 on page 180.
« Last Edit: Nov 29th, 2008, 5:31am by immanuel78 » IP Logged
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board