value of complex integral $\int_C {\frac{e^{-z}}{z+1}}dz$ over the circle $|Z| = 1/2$

Question

As per my understanding, $f(z)= {\frac{e^{-z}}{z+1}} $ is not analytic at $z=0$, so we cannot apply Cauchy's Fundamental Theorem. Which states " if $f(z)$ is analytic at all points within and on the closed contour C, then $\int_C f(z) dz = 0$"

But the answer provided is $0$ by applying Cauchy's Fundamental Theorem. What am i missing ?

Answer 1

Clearly, $f(z)$ is analytic at $0$, as you can expand it in Taylor's series around zero and it will converge for $|z|<1$. It is not analytic at $z=-1$, but that will not create problem as it is outside the domain.