# Solutions to $(z+1)^n = z^n$ using conformal maps.

I'm doing a homework problem where I have to find all roots of $(z+1)^7 - (z)^7 = 0$ using the roots of unity for $z^7$

I noticed that if $a$ is a root of unity for $z^7$, then $1/(a-1)$ maps the roots of $z^7$ to the roots of $(z+1)^7 - (z)^7$. I also noticed that this works for any power $n$, not just $n=7$. Similarly if we are solving $(z+c)^n - (z)^n = 0$, the solution appears to be $c/(a-1)$ where $a$ is the roots of unity for $n$.

furthermore, it seems for a polynomial $p(x)$, there is a conformal mapping between the roots of $p(x+1) - p(x)$ and the roots of $p(x)$

I discovered all of this while on mathematica and would appreciate an explanation to what is going on?

FYI this is my first post here!!

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You're looking for the roots of: $$\left(\frac{z+c}{z}\right)^n=1$$ Say $a^n=1$, then $(z+c)/z = a$, or: $$z(a-1)=c \ \to \ z = \frac{c}{a-1}$$