How can I find a contour integral over a curve that is a union of two circles?

Question

I have the problem: let C be the union of two circles of radius 1 and radius 2 centered at 0, oriented in opposite directions. Compute $\int_C \frac{z}{1-e^z}dz$ In this case, because the circles are oriented in opposite directions, does the inner one cancel out some of the outer one? So am I only concerned with the area in between the two curves? And if this is the case, since the issue with the function is at the point z=0, does it not matter? I've only computed contour integrals over single curves with possible issue points inside, so I am unsure what to do when my curve is actually two curves.

Answer 1

We are not talking about the union of the two circles, but about the chain $\gamma$ which is the formal sum of the circle of radius $2$ counterclockwise and the circle of radius $1$ clockwise: $$\gamma=\partial(D_2)-\partial(D_1)\ .$$ This chain $\gamma$ is the boundary cycle $\partial\Omega$ of the region $\Omega=\{z\,|\, 1<|z|<2\}$. For any function $f$ analytic in a neighborhood of $\bar\Omega$ Cauchy's theorem gives $$\int_{\partial(D_2)} f(z)\>dz-\int_{\partial(D_1)} f(z)\>dz=\int_{\partial\Omega} f(z)\>dz=0\ .$$

Answer 2

Yes, with one circle inside the other, the only issue is singularities in the region between the two circles. Here there is no singularity between the two circles so the integral is 0.