Question

Let C be any simple closed contour, described in the positive sense in the z plane and write g(z)= Integral over C of (s^3+2s)/(s−z)^3 and show that g(z)=6*pi*i*z when z is inside C and g(z)=0 when z is outside

Answer 1

what is s?

Answer 2

oh, s is just a random variable, like x or y

Answer 3

but it is important to know where it is relative to C

Answer 4

if it is outside, then for z inside C function would be anlytic, so it's integral over any closed path is 0

Answer 5

I think that something is missing here

Answer 6

ok, this may sound skinda weird, but i'll try. so treat the equation more like this:
g(z0)= Integral over C of (z^3+2z)/(z−z0)^3. z is like, the function, whereas z0 is the point of concern. I have the first part of the answer, the 6*pi*z, that's easy, you just use the cauchy-integral formula. however, i dont know how to solve the second part, for outside the circle

Answer 7

Hmm, why you didn't write it like this from beggining.... Notice that if z0 is outside the countour, then integrand function is analytic inside it, so it's integral =0 by Cauchy's theorem. If you don't believ that try to solve it for circular countour around z_0, which you can do because of the homotopy