The Cauchy Integral Formula

  • Thread starter AlBell
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  • #1
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Homework Statement



Use Cauchy's integral formula to evaluate
34t8wo3.png
when
a) C is the unit circle
b) c is the circle mod(Z)=2

Homework Equations



I know the integral formula is
14mtco.png



The Attempt at a Solution


for the unit circle I was attempting F(z)=sin(z) and Z0=∏/2, which would give a solution of 2∏i, however if this is the correct method I am unsure how to modify it for a larger unit circle as I thought the final result was independent of radius
 

Answers and Replies

  • #2
Office_Shredder
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The integral formula requires the point z0 to be contained inside of the curve gamma that you are integrating around. Draw some pictures and you should see the difference between the two curves they are asking you to integrate on
 
  • #3
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Ah so the unit circle wouldn't actually contain the point pi/2 whereas the circle mod(z)=2 would?
 
  • #4
Office_Shredder
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That's right. So in the unit circle case you need to figure out something else that lets you calculate the integral
 
  • #5
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Can I then use the integral theorem that says it will equal 0?
 
  • #6
Office_Shredder
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That will work
 
  • #7
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I'm a bit confused again, sorry!
I thought that the z-Pi/2 on the denominator of the integral means we just shift the origin of the circle to a new position?
 

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