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        general >> complex analysis >> sequence
        
(Message started by: Alex on Jun 17^th, 2004, 3:43pm)

Title: sequence
Post by Alex on Jun 17^th, 2004, 3:43pm

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.

Title: Re: sequence
Post by Icarus on Jun 23^rd, 2004, 7:28pm

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 - a_k), and let Q_i = Q(x)/(x - a_i), then P_k(x) = [sum]_i b_iQ_i(x)/Q_i(a_i)

Now let i [to] [infty] ...

Title: Re: sequence
Post by puzzlecracker on Nov 29^th, 2004, 10:10pm

Alex are you Russian and where do you study?

Title: Re: sequence
Post by Icarus on Dec 5^th, 2004, 7:07pm

Out of curiousity, why do you Alex might be Russian?