David Ullrich : Please if you would help with a previous question

From: Brett (BrettMandich222_at_hotmail.com)
Date: 01/20/05

  • Next message: The World Wide Wade: "Re: Easy probability question" 
    Date: Wed, 19 Jan 2005 22:22:30 -0500

    Dear David Ullrich,

    I asked you about a question before about how to find a merormorphic
    function on the complex plane with simple poles at n = 1,2,3,... and with
    principal part

    sqrt(n)/(z-n) n = 1,2,3,....

    The key to the problem you said was

    "For example, it's easy to see (how?) that
    there exist polynomials P_n such that

      |P_n(z) + sqrt(n)/(z-n)| < 1/2^n

    for all z with |z| < n-1."

    I agree if I can solve this, then the problem is solved.

    Please tell me if the following is correct :

    First Expand sqrt(n)/(z-n) as a power series as follows :

    sqrt(n)/(z-n) = (-sqrt(n) / n )*(1/(1-z/n)) = (-sqrt(n)/n) * sum (z/n)^n

    where we need |z/n| < 1 for the sum to converge. Now, I believe that our
    P_n(z) will be partial sums of the last infinite sum where |z| < n-1. We
    need |z| < n-1 here because then |z|/n will be less than 1 in the infinite
    sum so we can eventually get |P_n(z) + sqrt(n)/(z-n)| < 1/2^n. Do I have
    the correct way to solve this?

    Thank you for your help,

    Brett

  • Next message: The World Wide Wade: "Re: Easy probability question"

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