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Determing where function is differentiable (Complex Analysis)

  1. Oct 4, 2007 #1
    1. The problem statement, all variables and given/known data
    Determine where the function f has a derivative, as a function of a complex variable:

    f(x +iy) = 1/(x+i3y)


    3. The attempt at a solution

    I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?
     
  2. jcsd
  3. Oct 4, 2007 #2

    Dick

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    No. Where Cauchy-Riemann is satisfied, the function is differentiable. There could be some places. Are there?
     
  4. Oct 4, 2007 #3
    Hmm I was thinking, maybe I could take the partials of U and V (Cauchy-Riemann) equate them, and find out what x,y have to be in order for the equations to be satisfied. Am I on the right track?
     
  5. Oct 4, 2007 #4

    Dick

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    Yes, if CR is satisfied at a point, then the function is differentiable at that point. That's where CR comes from.
     
    Last edited: Oct 4, 2007
  6. Oct 4, 2007 #5
    Alright, so doing the partials and equating I get

    x = 3y and
    xy = 3xy

    But, Im a little confused on what to do from here.

    Thanks for the help
     
  7. Oct 4, 2007 #6
    Thinking about it a little more, would this mean when x = 3y, the function is differentiable and when either x=0, y=0, the function is differentiable? Or is my thinking wrong?
     
  8. Oct 4, 2007 #7

    Dick

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    IF that's right then putting the first equation into the second gives 3y^2=9y^2, so y=0. If y=0 then x=0. So differentiable only at 0+0i.
     
  9. Oct 4, 2007 #8
    Thanks for the help Dick!

    Just to clarify one more thing. Presuming that I did calculate the paritals right and that leads to 0 + i0, wouldnt that lead to the original function being undefined? As a result, wouldnt the function be differentiable no where?
     
  10. Oct 4, 2007 #9

    Dick

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    Very astute, scothoward. I was on my way to missing that.
     
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