Give an example of a function $f$ which has a simple pole at $z=i-3$ with residue $8(i-3)$ and an essential singularity at $z=i$ with residue $6$

Question

Give an example of a function $f$ which has a simple pole at $z=i-3$ with residue $8(i-3)$ and an essential singularity at $z=i$ with residue $6$.

I know that the function $-24/(z+3-i)$ would have a pole at $z=i-3$ with residue $8(i-3)$ but I cannot seem to make this function also have an essential singularity. Would making this function trigonometric help.

Answer 1

Hint. Try something of the form $$f(z)=e^{1/(z-i)^2}+\frac{A}{z-i}+\frac{B}{z-(i-3)}$$ where $A$ and $B$ are complex numbers to be found. The essential singularity is given by $e^{1/(z-i)^2}$ which has residue $0$ at $i$ and at $i-3$.

Answer 2

Hint:

1. $e^{1/(z-i)}=1+\frac1{z-i}+\frac1{2(z-i)^2}+\dots$ has an essential singularity at $z=i$ with residue $1$.

2. $\frac1{z+3-i}$ has a simple pole at $z=i-3$ with residue $1$.