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Math Help - Complex analysis problem

  1. #1
    Junior Member
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    Complex analysis problem

    Hi! I need some help with the following problem:

    Let f(z) be analytic in |z|< 1, and f(0) = 0. Define F(z)=f(z)/z for every z in 0<|z|<1. What value can we give to F(0) for F(z) to be analytic in |z|<1? (Hint: Apply Cauchy's integral formula for derivatives to F(z) in |z|=r < 1. Then show that the resulting function, analytic in |z|< 1, concurs with F in 0<|z|<r. Use partial fractions.)

    So, Cauchy's integral formula for derivatives states that

    f^{(n)}(a)=\frac{n!}{2\pi i}\int_{\gamma}\frac{f(z)}{(z-a)^{n+1}}dz, n=1,2...

    But I'm not sure how to apply it to F(z).

    Any help will be appreciated.
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  2. #2
    MHF Contributor
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    Re: Complex analysis problem

    Hey Cesc1.

    Setting a = 0 gives f(z) / z^(n+1) given the line integral region of y (gamma). If a function is analytic in the complex plane then all of its derivatives exist by default in the region that the function is itself analytic.

    This means if f(z) is analytic in a region then its derivatives are also analytic in the same region (i.e. the f^n(a)).

    So you need to break up f(z)/z^(n+1) into f(z)/z and 1/z^(n+1) and show that F(z) is analytic.

    You have already shown that f^n(0) exists for all values of n (since 0 is in the region |z| < 1) so you need to use F(z) * 1/z^n in partial fraction decomposition within the integral.
    Thanks from Cesc1
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