Difficult Algebra Problem in Conformal Mapping

Question

I've got a question regarding algebra where I simply need to rearrange and solve for another variable. A little background: the expression comes from conformal mapping of the unit circle (in the $z$ plane) into $n$ number of cuts (located on the $\omega$ plane). The equation involves $\omega$ as a function of $z$ and $n$ where $n$ is an integer. I'm trying to solve for $z$ so that I can go from $n$ number of cuts into a circle. The equation is as follows:

$\omega=\frac{1}{4^{\frac{1}{n}}}\frac{\left(z^n+1\right)^\frac{2}{n}}{z}$

I've tried to rearrange the expression for $z$ several times myself but cannot get the correct mapping so I must be doing something wrong. Any ideas would be much appreciated.

Thank you for your help

Answer 1

Here is a naive way; I don't know if there are any better options. And perhaps it's something you've already tried, you didn't give many details.

• Multiply both sides by $4^{1/n}z$:

$$4^{1/n}z\omega = (z^n + 1)^{2/n}$$

• Raise both sides to the $n$ power and expand the RHS:

$$4z^{n}\omega^{n} = (z^n + 1)^2 = z^{2n} + 2z^n + 1$$

• Set equal to $0$, and notice it's quadratic in $z^n$:

\begin{align*} 0 &= z^{2n} + (2 - 4\omega^n)z^n + 1 \\ 0 &= (z^n)^2 + (2 - 4\omega^n)z^n + 1 \end{align*}

Now you can use the quadratic formula (with $a = 1,~b=2 - 4\omega^n,~c=1$) to solve for $z^n$ and take $n\text{th}$ roots for $z$.

It will not be very pretty (although the discriminant isn't too bad), but the expression for $\omega$ wasn't particularly good-looking either.