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Complex Analysis: prove the function is entire

  1. Feb 3, 2012 #1
    Hey guys, i just started a complex analysis course this semester and we just went over CR-equations, and various ways to show that a function is holomorphic. I'm a bit stuck on this one homework question where we have to prove the function is entire.

    1. The problem statement, all variables and given/known data

    so we have f(z)=(z^2-2)(e^-x)(e^-iy)

    3. The attempt at a solution
    First I tried expanding the e^-iy out to cos(-y)+isin(-y) and tried to multiply it out to get a u and a v. That required a lot of tedious algebraic manipulation for one of the earlier problems in the pset for this prof so then I tried looking for a more elegant solution. I turned (e^-x)(e^-iy) into e^(-z), and tried to show that the partial derivative with respect too the conjugate of z = 0, but I'm still not to clear on that whole method. Was wondering if anyone can point me along the right direction/explain something maybe I don't understand.

    Thanks
     
  2. jcsd
  3. Feb 3, 2012 #2

    lanedance

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    I think multplying out as you started with is not a bad way to approach this
     
  4. Feb 3, 2012 #3
    Hmmm, maybe I did not put enough work in that direction before trying other stuff. I'll work on it and get back.

    EDIT: I was wondering also, for the criteria that partial(f)/partial(zbar)=0, is zbar treated like any other varaible? In other words, since I can express the first function entirely in terms of z, can I say that partial(f)/partial(zbar) = 0 and so say that they are holomorphic?
     
    Last edited: Feb 3, 2012
  5. Feb 3, 2012 #4

    Dick

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    Sure, your function is e^(-z)*(z^2-2). It seems a little painful to show CR applies directly. You don't know things like the product of analytic functions is analytic??
     
  6. Feb 3, 2012 #5
    mmm, i think this method and the partial(f)/partial(zbar)=0 method work to make this problem very easy. We never proved product of analytic functions is analytic, maybe he expected us to know this from analysis I. I skipped it because I wouldn't have otherwise been able to fit complex analysis into my 4 years here :( Been keeping up fine teaching myself stuff I'm not familiar with though.

    lol im an idiot, I wasn't understanding the conjugate of z equation properly. Looked back at the derivation and i get it now.
     
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