
#1
Mar2812, 08:39 AM

P: 11

1. The problem statement, all variables and given/known data
Use Cauchy's integral formula to evaluate when a) C is the unit circle b) c is the circle mod(Z)=2 2. Relevant equations I know the integral formula is 3. The attempt at a solution for the unit circle I was attempting F(z)=sin(z) and Z_{0}=∏/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 



#2
Mar2812, 08:49 AM

Mentor
P: 4,499

The integral formula requires the point z_{0} 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
Mar2812, 08:57 AM

P: 11

Ah so the unit circle wouldn't actually contain the point pi/2 whereas the circle mod(z)=2 would?




#4
Mar2812, 08:58 AM

Mentor
P: 4,499

The Cauchy Integral Formula
That's right. So in the unit circle case you need to figure out something else that lets you calculate the integral




#5
Mar2812, 09:15 AM

P: 11

Can I then use the integral theorem that says it will equal 0?




#6
Mar2812, 09:24 AM

Mentor
P: 4,499

That will work




#7
Mar3112, 05:51 AM

P: 11

I'm a bit confused again, sorry!
I thought that the zPi/2 on the denominator of the integral means we just shift the origin of the circle to a new position? 


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