wu :: forums « wu :: forums - the Extended Hurwitz's Theorem » Welcome, Guest. Please Login or Register. Feb 8th, 2023, 7:21pm RIDDLES SITE WRITE MATH! Home Help Search Members Login Register wu :: forums  general  complex analysis (Moderators: Grimbal, SMQ, william wu, ThudnBlunder, Icarus, Eigenray, towr)  the Extended Hurwitz's Theorem « Previous topic | Next topic » Author Topic: the Extended Hurwitz's Theorem  (Read 10816 times)
immanuel78
Newbie   Gender: Posts: 23 the Extended Hurwitz's Theorem   « on: Oct 26th, 2008, 7:49am » Quote 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: Posts: 500 Re: the Extended Hurwitz's Theorem   « Reply #1 on: Nov 3rd, 2008, 8:09pm » Quote Modify

you mention before regarding Hardy's Theorem  came form
Conway (actually its problem from the book?). IP Logged

Regards,
Michael Dagg
immanuel78
Newbie   Gender: Posts: 23 Re: the Extended Hurwitz's Theorem   « Reply #2 on: Nov 29th, 2008, 5:26am » Quote 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

 Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth => complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »