Evaluating Complex Integrals Using Cauchy's Integral Formula

[AlBell]

Homework Statement

Use Cauchy's integral formula to evaluate

when
a) C is the unit circle
b) c is the circle mod(Z)=2

Homework Equations

I know the integral formula is

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 let's 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?