I have to provide a function $f(z)$ which is analytic in C except for (i) at a single pole at $z = 3$, (ii) a removable singularity at $z=0$ and (iii) an essential singularity at $z=-1$.
I've defined $f(z) = f_i +f_{ii} + f_{iii}$.
Where $f_{i}(z) = \frac{z+1}{z+3}$, $f_{ii}(z) = \frac{1}{z}$, and $f_{iii}(z) = sin\left(\frac{7}{z+1}\right)$.
I have referred to this site.
How about $$f(z)=\frac{z+1}{z-3} +\frac{\sin z}{z}+e^{\frac{1}{z+1}}.$$