How can I find the real and imaginary parts of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$? By using $x = \sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ , I can find a quadratic equation, but how can I separate the real and imaginary parts?
Edit:
My question was how to find the real and imaginary parts of the complex number $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$. 
After solving the quadratic equation we'll get the solution - $x_{1,2} = \frac{1}{2}\pm \frac{\sqrt{1+4i}}{2}$. Now the prblem is to find the real and imaginary parts of $\sqrt{1+4i}$. 
Sorry for the confusion.

x = $\sqrt{(i+\sqrt{(i+\dots)})}$ = $\sqrt{(i+x)}$
$x^2 - x - i = 0$
Solving for x we get,
$x_{1,2} = \frac{1}{2}\pm \frac{\sqrt{1+4i}}{2}$
Now,
$a + ib =\sqrt{1+4i}$
$\implies a^2 - b^2 + 2iab = 1+4i$
So we have, $a^2-b^2 = 1$ and 2ab = 4.
Solving for a and b we can get
$a = \sqrt(\frac{\sqrt{17}+1}{2})$ and  $b = \sqrt(\frac{\sqrt{17}-1}{2})$.
So the final answer is-
$x = \frac{1}{2} + (\frac{\sqrt{\sqrt{17}+1}}{2\sqrt{2}}) + i (\frac{\sqrt{\sqrt{17}-1}}{2\sqrt{2}})$. 
