How Can the Residue Theorem Be Applied to Prove This Integral?

[de1irious]

Hi, so suppose f(z) is a complex function analytic on {z|1<|z|} (outside the unit circle). Also, we know that limit as z-->infinity of zf(z) = A. Now I need to show that for any circle C_{R} centered at origin with radius R>1 and counterclockwise orientation, that

\oint f(z)dz = 2\pi iA

Any ideas? I'm trying to use Cauchy integral theorem somehow but it's not working.

 

[Pere Callahan]

Are you familiar with the Riemann sphere?

In this picture, every closed curve in C can be considered to go around infinity (which is just one point on the sphere). If you choose a circle P with radius larger than R then your function is analytic in the connected component of C\P which contains infinity.

The residue of your function at infinity is A so the Residue Theorem implies your assertion at once.