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Power series for complex function

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the power series for the function
    f(z) = (1-z)^-m

    Hint: Differentiation gives:
    f'(z) = m(1-z)^m-1
    = m(1-z)^-1.f(z)

    or:
    zf'(z) + mf(z) = f'(z)

    Use the formula for differentiation of power series to determine the coefficients of the power series for f.


    2. Relevant equations



    3. The attempt at a solution

    Hi, here's my attempt so far:

    From real analysis,
    (1-x)^-1 = Σ(a_n).x^n, where the sum is from n=0 to infinity

    Differentiating:f'(x) (1-x)^-2 = Σ(a_n).n.(x^n-1) , from n=0 to infinity

    and so on until we get that the j-th derivative

    i.e. f^j.(x) = Σa_n(n C j).(x^n-j), where (n C j) is "n choose j" (binomial coefficient.

    Thus, substituting z for x and m for j,

    (1-x)^-m = Σa_n(n C m).(x^n-m), sum from 0 to infinity.


    ---

    I don't think this is correct; it seems too straightforward and I haven't used the hint.
    Can someone please point me in the right direction?

    Thanks for any help!
     
  2. jcsd
  3. Nov 17, 2009 #2

    turin

    User Avatar
    Homework Helper

    ... not to mention that you still have an infinite number of undetermined coefficients.

    They give you a first order ODE. Solve it using power series.
     
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