Author |
Topic: sequence (Read 2548 times) |
|
Alex
Guest
|
Please help me with this problem
Let {a_n} and {b_n} be sequence of complex numbers. Assume that {a_n}
has no accumulation point. Prove that there exists a holomorphic
function f:C->C such that f(a_n)=b_n
Thank you.
|
|
 IP Logged |
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender: 
Posts: 4863
|
 |
Re: sequence
« Reply #1 on: Jun 23rd, 2004, 7:28pm » |
 Quote  Modify
|
Start with a finite version: Find a polynomial P1 of minimal degree such that P(a1) = b1. The answer is trivial: P1(x)=b1. Now find P2, which works for a1 and a2: A trick for this one is two start off with A(x - a1) + B(x - a2). At a1, the first term is zero, so all contribution comes from the second term - yielding B(a1 - a2). Choosing B to be b1/(a1 - a2) yields the value we want. Similarly, A is chosen to be b2/(a2-a1).
So P2(x) = (b1(x - a2) - b2(x - a1))/(a1-a2)
If we let Q(x) = (x - a1)...(x - ak), and let Qi = Q(x)/(x - ai), then Pk(x) = [sum]i biQi(x)/Qi(ai)
Now let i [to] [infty] ...
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
puzzlecracker
Senior Riddler
Men have become the tools of their tools
Gender:
Posts: 319
|
|
Re: sequence
« Reply #2 on: Nov 29th, 2004, 10:10pm » |
Quote Modify
|
Alex are you Russian and where do you study?
|
|
IP Logged |
While we are postponing, life speeds by
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: sequence
« Reply #3 on: Dec 5th, 2004, 7:07pm » |
Quote Modify
|
Out of curiousity, why do you Alex might be Russian?
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print
Remove