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Cauchys theorem

  1. Oct 11, 2011 #1
    Hey,

    This is just a small question about Cauchys theorem.

    If there is a function f(z) such that int f(z)dz = 0 can you conclude f is analytic in and on the region of integration?

    What I mean is can you work the theorem in reverse?

    For example if the above is true over a region C which is a simple closed curve, is f(z) analytic both inside and on C?
     
  2. jcsd
  3. Oct 11, 2011 #2

    mathwonk

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    check our morera's theorem.
     
  4. Oct 12, 2011 #3

    HallsofIvy

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    That depends on exactly what you mean by "int f(z) dz". If you mean simply that [itex]\oint f(z)dz= 0[/itex] for some closed path, no. For example, f(z)= 1 for Im(z)> 0, f(z)= -1 for Im(z)< 0, the integral around any circle centered on the origin is 0 but that function is not analytic. Morera's theorem, that mathwonk suggests, says that if the integral around every closed path in a region is 0, then the function is analytic in that region.
     
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