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Homework Help: Complex power series question

  1. May 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Suppose that f(z) = ∑a_j.z^j for all complex z, the sum goes from j=0 to infinity.

    (a) Find the power series expansion for f'
    (b) Where does it converge?
    (c) Find the power series expansion for f^2
    (d) Where does it converge?
    (e) Suppose that f'(x)^2 + f(x)^2 = 1, f(0) = 0, f'(0) = 1

    Find a_0, a_1, a_2, a_3, a_4, a_5

    2. Relevant equations

    3. The attempt at a solution

    Hi everyone, here's my attempt. I'm not sure if it's right though, so I would really appreciate it if you could please take a look at it:

    (a) f'(z) = ∑j.a_j.z^(j-1), sum from 0 to infinity

    = ∑j.a_j.z^(j-1), sum from 1 to infinity

    = ∑(j+1).a_(j+1).z^j, sum from 0 to infinity

    (b) We don't what the values of a_j are, so we can only say with definity that it converges for z = 0

    (c) f^2 = (a_0)^2 + (a_0.a_1)z + (a_0.a_2)z^2 + ...
    + (a_1.a_0)z + (a_1.a_1)z^2 + ... etc.

    (d) Again, we can only say it converges for definite where z = 0.

    (e) f(0) = 0
    a_0 + a_1(0) + a_2(0) +... = 0

    i.e. a_0 = 0

    f'(0) = 1
    a_1 + 2.a_2.z + 3.a_3.z^2 +.... = 1

    The fact it is equal to 1 means it converges, but, as above, this can only happen if z = 0
    i.e. a_1 = 1

    Calculate f'(x)^2 = 1 + z(4.a_2) + z^2(6.a_3 + 4.a_2.a_2) + ...

    and f(x)^2 = z^2(a_1.a_1) + z^3(2.a_1.a_2) + ...

    The coefficients of the z terms in f'(x)^2 must be equal to minus the coefficients of the corresponding z terms in f(x)^2.

    So we find that:

    a_2 = 0
    a_3 = -1/6
    a_4 = 0
    a_5 = 1/120

    Regarding the convergence in (b) and (d), am I correct in what I say, in that there is no way of knowing what the a_j's are, so the only way we can know the series converge is when z = 0?

    Also, have I multiplied the series correctly?

    I'm sure I must have done something wrong, as it's a question from a past paper in my complex analysis class, but I don't seem to have used any complex analysis here...

    Thanks for any help!
  2. jcsd
  3. May 10, 2010 #2


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    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    You can do better than that. Try applying the ratio test and use the fact the original series converges everywhere.
    You should collect terms. There's a pattern to the coefficient of zn.
    Again, you should be able to do better than that. Do you know any theorems about convergence of products of series?
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