# How to go about solving $((1+iz)/(1-iz))^4 = 1/2 + i\sqrt3/2$?

I have problem solving this equation: $$\left(\frac{1+iz}{1-iz}\right)^4 = \frac12 + i {\sqrt{3}\over 2}$$ I know how to solve equations that are like: $$w^4 = \frac12 + i {\sqrt{3}\over 2}$$ And I have solved it to: $$w = \cos(-\frac{\pi}{12} + \frac{\pi k}{2})) + i\sin(-\frac{\pi}{12} + \frac{\pi k}{2}))$$ But now is: $$w = \frac{1+iz}{1-iz}$$ How does one get the complex z? Or am I solving it wrong?

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Please add appropriate MathJax code so your question is understandable. – Parth Kohli Sep 23 '12 at 14:49
@ParthKohli: Yeah, I'm trying to get it to work. :) – Curtain Sep 23 '12 at 14:50
Thanks for the edit. – Curtain Sep 23 '12 at 14:52

$$w=\frac{1+\mathrm iz}{1-\mathrm iz}\iff z=\mathrm i\cdot\frac{1-w}{1+w}$$ Edit: On the road are the identities $(1-\mathrm iz)\cdot w=1+\mathrm iz$ and $1-w=-\mathrm i\cdot(1+w)\cdot z$.
$$w=\frac{1+iz}{1-iz}$$ First, multiply both sides by $1-iz$: $$w(1-iz) = 1+iz$$ Expand the left side: $$w-wiz = 1+iz$$ Put all terms involving $z$ on one side and those not involving $z$ on the other side: $$w-1=iz+wiz$$ Factor $$w-1 = iz(1+w)$$ Divide both sides by $i(1+w)$: $$z= \frac{w-1}{i(w+1)}$$ Multiply the numerator and denominator by the conjugate, $-i$: $$z = -i\frac{w-1}{w+1} = i\frac{1-w}{1+w}.$$