Geometrical interpretation of the complex map $ f(z)=\frac{1}{(z-i)^n} $ In a complex variable class, a professor asked us the following.

Describe the function $$ f(z)=\frac{1}{(z-i)^n} $$ geometrically.


I tried writing $z=x+iy$ to see what the function does to lines and circles. Initially I got
\begin{equation}
f(z)=\frac{(\bar{z}+i)^n}{\lvert z-i\rvert^{2n}}=\frac{1}{(x^2+(y-1)^2)^n}\sum_{k=1}^n\binom{n}{k}x^{n-k}(1-y)^ki^k
\end{equation}
but it got me nowhere. I then tried restricting the domain to the real line and obtained
\begin{align}
f(x+0i)=\frac{1}{(x^2+1)^n}\sum_{k=1}^n\binom{n}{k}x^{n-k}i^k&=\frac{1}{(x^2+1)^n}\sum_{\quad \,\,k=1\\k\equiv 0,2\operatorname{mod}4}^n\binom{n}{k}x^{n-k}(-1)^k\\ &\quad+\frac{i}{(x^2+1)^n}\sum_{\quad \,\,k=1\\k\equiv 1,3\operatorname{mod}4}^n\binom{n}{k}x^{n-k}(-1)^k
\end{align}
And now I think the problem might not have a solution. I'm starting to think there's no clear way to describe this geometrically. Any ideas are gladly appreciated.
 A: As you were told, it is geometric interpretation that  you have been asked.
It is best to take an example that you should follow by drawing in the $OJI$ plane($O=0, J=1$ and $I=i$):
Let $n=6$ and $z=A=\frac12+\frac12i$.
$z-i=\frac12+\frac12i-i=\frac12-\frac12i=A-I=\vec{IA}=\vec{OA'}=A'-O=A'$, with $A'=\frac12-\frac12i=\frac{\sqrt2}{2}e^{-i\frac{\pi}{4}}$
Then $\frac{1}{z-i}=\frac{1}{\frac{\sqrt2}{2}e^{-i\frac{\pi}{4}}}=\frac{1}{\frac{\sqrt2}{2}}e^{i\frac{\pi}{4}}=\sqrt2e^{i\frac{\pi}{4}}$(observe the effect on the argument and the modulus.) Then I built the example for this, everything works fine, but it remains anecdotal for the explanation: $\frac{1}{z-i}=1+i$, which is a point that we denote $A'':=1+i=\sqrt2e^{i\frac{\pi}{4}}$
Then $\frac{1}{(z-i)^n}=(\frac{1}{(z-i)})^n=A''^6=(\sqrt2e^{i\frac{\pi}{4}})^6=(\sqrt2)^6(e^{i\frac{\pi}{4}})^6=8e^{3i\frac{\pi}{2}}=-8i$(observe the effect on the argument and the modulus.)
This type of explanation, even a mathematician like Adrien Douady did not shy away from giving them in his famous film "the dynamics of the rabbit" (sorry, it's in French)
