Cauchy Integralformula comprehension question

Question

So, I saw the following step in a lecture$ \int_{|z|=3}^{} \frac{e^{-z}}{(z+2)^3}dz=\frac{2 \pi i}{2}e^2 \ $ with the cauchy integralformula where |z|=3 is the circle around 0 with radius 3. Now my question, doesn't the formula require the function to be holomorphic on the circle and on the entire set enclosed by the circle? If so, shouldn't the formula not be applicable here since there is a singularity inside the circle at -2? Or is there an implicit step here that circumvents the singularity?

Answer 1

It is the generalized Cauchy integral formula:

$$f^{(n)}(w) = \frac{n!}{2\pi i}\int_{\Gamma} \frac{f(\xi)}{(\xi-w)^{n+1}} d\xi$$

for the holomorphic function $f(z) = e^{-z}$, $\Gamma$ the circle $|z|=3$, and $n=2$.

Now, for your question

Now my question, doesn't the formula require the function to be holomorphic on the circle and on the entire set enclosed by the circle?

the answer is Yes and something more. More precisely, if $f:D \subseteq \mathbb{C} \rightarrow \mathbb{C}$, $\Gamma \subseteq D$ is the image of a closed simple curve and $K$ is the interior of $\Gamma$, then in order to use the (generalized) Cauchy integral formula one has to verify that there exists some open set $U\subseteq D$ on which $f$ is holomorphic such that $\overline{K} \subseteq U$.