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Writing a function in u+iv form

  1. Jul 13, 2012 #1
    This is related to another post of mine. How would you go about writing [itex]\frac{1}{e^{z}-1}[/itex] in u+iv form? Usually multiplying through with the complex conjugate gives you the desired form, but here I'm not sure that it works. Any suggestions?
     
  2. jcsd
  3. Jul 13, 2012 #2

    haruspex

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    Write z = x+iy. Massage the denominator into u+iv form, then multiply top and bottom by conjugate.
     
  4. Jul 13, 2012 #3

    With [itex]\,z=x+iy\,\,,\,x,y,\in \Bbb R\,\,\,and\,\,\,e^{ix}=\cos x+i\sin x[/itex] :

    [tex]\frac{1}{e^z-1}=\frac{1}{e^x\cos y -1+ie^x\sin y}=\frac{e^x\cos y-1-e^xi\sin y}{e^{2x}-2e^x\cos y+1}=\frac{e^x\cos y-1}{e^{2x}-2e^x\cos y+1}-\frac{e^x\sin y}{e^{2x}-2e^x\cos y+1}\,i[/tex]

    Piece of cake (pant,pant!)

    DonAntonio
     
  5. Jul 13, 2012 #4

    Mark44

    Staff: Mentor

    Write ez as ex + iy = exeiy = ex(cosy + i siny). Your denominator is this expression, minus 1.

    It's slightly messy, but you can rationalize the denominator by multiplying by the conjugate over itself.
     
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