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Even function has a Laurent decomposition of even functions and even powers of z

  1. Jul 26, 2009 #1
    Hi,

    if let's say that there's an even function f(z) then how do we know if its Laurent decomposition (i.e. f(z) = f0(z) + f1(z) ) will be even functions and have even powers of z?

    Any help will be greatly appreciated.
     
  2. jcsd
  3. Jul 26, 2009 #2

    Hurkyl

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    Take a generic Laurent series and compute it's even and odd parts. What do you see?

    (You know a formula for the even and odd part of any function, right?)
     
  4. Jul 26, 2009 #3
    i know that for even function f(z) = f(-z) which means that the sum from n=0 to infinity of a_n z^n is equals to the sum from n=0 to infinity of a_n (-z)^n.
    (note: a_n is the coefficient of the series)

    Then this will give me an equation of a_n = a_n (-1)^n ---> conclusion: a_n can't be equal to 0, am I right?

    But then after this I don't know what else to do in order to prove that f0(z) and f1(z) are even functions that only has powers of z.

    Thank you so much
     
  5. Jul 26, 2009 #4
    Must be Laurent series in powers of [itex]z[/itex].
     
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