I have complex analysis exam next week and one of the main question is to solve some integrals with the Cauchy formula which is $$f(z)=\frac{1}{2\pi i}\int \frac{f(w)}{w-z}dx$$ and I can solve a big part of the exercise that I'm asked to do but I have a problem regarding the domain.
If I have $\displaystyle \int $ in a domain like $|z|=3$, I can say whether a singularity belongs or not to the domain. However, when I have domain that do not have equations or inequalities I can not understand it.
For example if the domain is $|3e^i(^t)|$ which is if i'm not wrong a circumference with center in $2$, but does that mean that the singularity have to be inside?
Another question: if a singularity is not in the domain, what do I have to do? Cause my teacher sometimes says that "since the singularity does not belong to the domain then the integral is $0$ but sometìmes proceeds to find a similar way to express the integral and then finds an answer, should the integral be $0$ when the function is analytic?
You have the wrong formula in the post. The formula should be $$f(z)=\frac1{2\pi{i}}\oint_{\gamma}\frac{f(w)}{w-z}\,\mathrm{d}w,$$ where $\gamma$ is a closed contour in $U,$ the domain of $f,$ such that the region $D$ enclosed by $\gamma$ contains $z.$ It is very important that $f$ be holomorphic in the region enclosed, for this formula to work, and it is necessary that there be no singularities in the region enclosed either. If the region does enclose singularities, then instead, $$\frac1{2\pi{i}}\oint_{\gamma}\frac{f(w)}{w-z}\,\mathrm{d}w=\sum_{m=0}^n\mathrm{I}(\gamma,c_m)\mathrm{Res}\left(\frac{f(w)}{w-z},c_m\right),$$ where the $c_m$ are the singularities enclosed, and $\mathrm{I}(\gamma,c_m)$ is the corresponding winding number.
I assume, from the context of your post, is that the contour given to you is a closed curve $\gamma$ as a function from $[0,2\pi]$ or $[-\pi,\pi]$ to $\mathbb{C},$ whose graph is $\gamma(t)=2+3e^{it},$ since you said the curve was a circumference with center $2.$ The domain of $f$ is then some open set $U$ that contains this circumference, and the region enclosed by this curve, $D,$ is the region where you need to check for singularities. The specifics of $U$ do not actually matter for the purpose of your exercise, only the region $D,$ which is the region enclosed by the given $\gamma,$ matters.