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Math Help - Complex variables

  1. #1
    Senior Member vincisonfire's Avatar
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    Complex variables

    Find all functions  f (z ) satisfying the following two conditions:
    (1) f (z ) is analytic in the disk |z - 1| < 1 .
    (2)  f ( \frac{n}{n + 1} ) = 1 - \frac{1}{2n^2 + 2n + 1} .
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  2. #2
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    Quote Originally Posted by vincisonfire View Post
    Find all functions  f (z ) satisfying the following two conditions:
    (1) f (z ) is analytic in the disk |z - 1| < 1 .
    (2)  f ( \frac{n}{n + 1} ) = 1 - \frac{1}{2n^2 + 2n + 1} .
    \frac1{2n^2+2n+1} = \frac1{(n+1)^2+n^2} = \frac{\frac1{(n+1)^2}}{1+\bigl(\frac n{n+1}\bigr)^2} = \frac{\bigl(1-\frac n{n+1}\bigr)^2}{1+\bigl(\frac n{n+1}\bigr)^2}<br />
, so you can take f(z) = 1 - \frac{(1-z)^2}{1+z^2}. (But could there be any other analytic functions taking those values at the points n/(n+1)?)
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Opalg View Post
    ... but could there be any other analytic functions taking those values at the points n/(n+1)?...
    In a problem i'm working about one important step is to demonstrate this lemma...

    Let be f(*) an analytic function whose value f_{n} are known for z=0,1,...,n, .... In this case, under certain conditions, there is only one analytic f(*) for which is f(n)= f_{n}.

    Does Opalg think that is an interesting question to be discussed in MHF?... if yes, in which section?...

    Kind regards

    \chi \sigma
    Last edited by chisigma; March 26th 2009 at 05:38 AM.
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  4. #4
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    Quote Originally Posted by chisigma View Post
    In a problem i'm working about one important step is to demonstrate this lemma...

    Let be f(*) an analytic function whose value f_{n} are known for z=0,1,...,n, .... In this case, under certain conditions, there is only one analytic f(*) for which is f(n)= f_{n}.

    Does Opalg think that is an interesting question to be discussed in MHF?... if yes, in which section?...
    Yes, that's a nice question. You could ask it as a new thread in this section.
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