Formulate a complex function with specific singularities

Question

I am trying to formulate a two complex functions with the following characteristics:

1. $f(z)$ has a zero of order 2 at $z=i$, a pole of order 3 at $z=-i$, and a pole of order 5 at $z=-(2+i)$.
2. $g(z)$ has a removable singularity at $z=0$, a pole of order 6 at $z=1$, and an essential singularity at $z=i$.

I'm thinking $f(z)=(z-i)^2(\frac{1}{(z+i)^3} + \frac{1}{(z+2+i)^5}$ and $g(z) = \frac{(z^2+z)e^\frac{1}{z-i}}{z(z-1)^6}$, but I'm unsure, mostly about $g(z)$, can anyone verify my answers?

Answer 1

For $g(z)$, I am not sure why you picked the $z^2+z$ to make the singularity at $0$ be removable, although it does work. I would have taken $$ g(z)=\frac{z}{z(z-1)^6}e^{\frac{1}{z-i}} $$ to really grind in the point that removable singularities can be real dumb.