Using the Cauchy Integral Formula compute the following integrals,where C is a circle of radius 2a centered at z=o, where 2a<pi
yes.Integrated around |z|=2a? If you write your integrand as f(z)/z (you figure out what f(z) would need to be) the Cauchy integral formula would tell you what the integral is in terms of f(0). Which would be very nice but the Cauchy integral formula doesn't apply because f(z)/z isn't holomorphic inside |z|=2a. Aside from a removable singularity at z=0 there's a pole at z=a. Are you sure you wrote the problem down correctly?
I have sperate the form like this:
The original problem is this:I really don't understand what you are saying there. Did you alter the original problem statement? If so, what was the original?
Ok, so you are really doing as a residue theorem problem, not just a Cauchy integral problem. In one of those integrals you should multiply by (z-0) and let z approach 0 and in the other one multiply by (z+a) and let z approach -a, right?
You should probably look it up. I don't necessarily explain things that well. The Cauchy Integral Theorem just says f(a)=(1/(2*pi*i)) times the contour integral f(z)/(z-a) over a circle where f(z) is holomorphic. The residue theorem is the obvious generalization of that to the case where you have multiple poles in a single domain and you cut out a circle around each one and add them up. Which is what you are doing.