The possible values of the $\frac{1}{2\pi i}\int_\gamma \frac{e^z}{(z-1)(z-2)}$

Question

If $\gamma$ is simple closed contour not passing through $1$ or $2$ then the integral $\frac{1}{2\pi i}\int_\gamma \frac{e^z}{(z-1)(z-2)}$ can have $7$ possible value. What are they? for each of these $7$ possible values,given an example of suitable $\gamma$

As if i take $\gamma:|z|=0.5$ then $z=1,2$ are out side so the value of integral is zero but how can we say remaining values integrals

Answer 1

So you know that $0$ is a possible value.

Now, if $\gamma$ is a closed simples path around $1$ but with $2$ on the outside, there will be $2$ possible values, one when the loop is traveled clockwise, and the other one when it is traveled in the opposite direction.

If $\gamma$ is a closed simples path around $2$ but with $1$ on the outside, you will get another $2$ values.

Finally, if both $1$ and $2$ are in the interior of the region bounded by the loop, you'll get the remaining $2$ values.

The specific values can be computed through the residue theorem ar applying partial fraction decomposition to $\frac1{(z-1)(z-2)}$ and then applying Cauchy's integral formula.