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Differentiating a complex power series

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data

    Say f(z) = Σ(z^n), with sum from 0 to infinity

    Then we can say f'(z) = Σn(z^n-1), with sum from 0 to infinity (i)

    = Σn(z^n-1), with sum from 1 to infinity (as the zero-th term is 0)

    = Σ(n+1)(z^n), with sum from 0 to infinity (ii)

    2. Relevant equations

    3. The attempt at a solution

    I am solving an ODE using power series, zf'(z) + af(z) = f'(z), so is it ok for me to sub in eq. (i) for the first f'(z) and eq. (ii) for the second? (This way all the terms will contain a z^n. Or is this cheating? If so, what can I do instead?)

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


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    Homework Helper

    Well, no. You aren't cheating. That seems like the right way to shift the sum. Carry on.
  4. Nov 18, 2009 #3
    Don't forget the constants in the sum, [tex] f(z) = \sum a_n z^n [/tex]
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