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Antiderive complex function f(z) and express as power series

  1. Aug 20, 2013 #1
    Let F(z) be the anti-derivative of the function f(z) = cos(z^3) with F(0) = 0. Express F(z) as a power series around z=0, giving both the first 3 non-zero terms and the general (nth) term.

    Hey guys really struggling with this integration and how to then express this as a power series. Any help at all would be greatly appreciated because I am getting nowhere!

    Thanks in advance!
     
  2. jcsd
  3. Aug 20, 2013 #2
    Start with the Taylor series of cosine. Is that enough to get you going?
     
  4. Aug 21, 2013 #3

    haruspex

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    You don't need to integrate anything. Just write out the general Taylor series for this unknown function F in the vicinity of 0.
     
  5. Aug 21, 2013 #4

    HallsofIvy

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    How is he going to find the anti-derivative without integrating?

    Jd303, Goa'uld's suggestion is best- write out the Taylor series for cos(z), replace "z" by z3, then integrate "term by term".
     
  6. Aug 21, 2013 #5
    Thanks guys I think I got it:
    Let u = z^3
    find Taylor series for f(u)
    sub back in z^3
    integrate term by term to find F(z)
     
  7. Aug 21, 2013 #6

    haruspex

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    You mean cos(u), right?
    Another way: F(z) = F(0) + z F'(0) + z2F"(0)/2! + ...
    You are given F(0), and you can substitute for F' etc. using f.
     
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