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Homework Help: Holomorphic functions

  1. Mar 8, 2010 #1
    I'm not too sure how to show this. Perhaps if I show d(ez)/dz = ez then does this conclude that ez is holomorphic on all of C?
     
    Last edited: Mar 8, 2010
  2. jcsd
  3. Mar 8, 2010 #2

    Dick

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    Why does d/dz(e^z)=e^z show e^z is holomorphic? Or at least, why are you saying you aren't sure this shows it's holomorphic?
     
  4. Mar 8, 2010 #3
    I suppose I should be using the cauchy riemann equations then?

    I was told not to use limits here so I didn't want to use the base definition of holomorphic-ness.
     
  5. Mar 8, 2010 #4
    Because in my notes he mentioned the power series
     
    Last edited: Mar 8, 2010
  6. Mar 8, 2010 #5

    Dick

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    Sure, use Cauchy-Riemann. That's easy enough. Then you'd want to show e^(-z) is also holomorphic. Do you know the sum of holomorphic functions is also holomorphic? If not then you could just directly show cosh(z) is holomorphic using CR.
     
  7. Mar 8, 2010 #6
    i tried to use C-R, but I'm unable to split it up into

    U(x,y) or V(x,y)

    using z=x+iy into cosh(z)
     
  8. Mar 8, 2010 #7

    Dick

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    How about e^z? Can you split that up?
     
  9. Mar 8, 2010 #8
    exeiy

    = ex(cosy + i.siny)

    =excosy + i.exsiny

    The C-R satisfy this and so it is holomorphic on all of C? so this would immediatly imply cosh(z) is entire?

    And I would check if it were true for e-z as well.
     
  10. Mar 8, 2010 #9

    Dick

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    Sure, check e^(-z) as well. It's almost the same thing. I'm a little puzzled why you can split e^(z) and e^(-z) up and not cosh(z). You already said cosh(z)=(e^z+e^(-z))/2.
     
  11. Mar 8, 2010 #10
    ah ye that would make more sense if i did it directly, thanks!
     
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