Cauchy integral theorem

1. Feb 28, 2008

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.

2. Feb 28, 2008

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.

Last edited: Feb 28, 2008