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Complex differentiable

  1. Nov 29, 2012 #1
    1. The problem statement, all variables and given/known data
    f(z)=z(bar(z))^2+2(bar(z))z^2 ,then calculate the total differential of f viewed as a map from R^2->R^2 . determine the points at which f is complex differentiable , is f holomorhpic anywhere????

    2. The attempt at a solution
    i did the first part and for secund part i use the Jocobian ,for if it is differentiable then if follow the cachy rieman equations which is 9x^2+3y^2=x^2+3y^2, and 6xy=-2xy the solution for the equation systems is x=o y is real , so it can be differentiable at (o,y) y is real number ,does it right??? if it is ,then How should i found holomorphic anywhere???
     
  2. jcsd
  3. Nov 29, 2012 #2

    Dick

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    That looks ok for differentiable. What's the definition of 'holomorphic'?
     
  4. Nov 29, 2012 #3
    f:k->C is holomorphic if f is complex differentiable at all point of the region K
    For the quesiton, for x=o so z=iy then the point can be differentiable is the line of imaginary axis ,for it is a line we cannnot define holomorphic here, then thats nowhere for f to be holomophic

    is my argument right??
     
  5. Nov 29, 2012 #4

    Dick

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    It's not very well worded. Sure, it's differentiable on a line. You'll want to explain why a 'line' isn't a 'region'. What property does a region have that a line doesn't?
     
  6. Nov 29, 2012 #5
    if i choose a point inside the region,then choose the very close small region around that point, in the point inside the small region should be diff. what i mean is the region should be open ,but the line we cannot find here, i didnt explain it in my post sorry
     
  7. Nov 29, 2012 #6

    Dick

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    Right. There's no point on the line that has a neighborhood of the point that's contained in the line. The line doesn't contain any open sets. It has empty interior.
     
    Last edited: Nov 29, 2012
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