The Cauchy Integral Formula

AlBell

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 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

Office_Shredder

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

AlBell

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

Office_Shredder

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

AlBell

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

Office_Shredder

That will work

AlBell

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?