I am trying to approximate the Stieltjes transform of the semicircle law. In particular, it is known that the Stieltjes transform m(z), for z in the upper half plane, is exactly
$$ m(z) = \frac{-z + \sqrt{z^2 - 4}}{2} $$ I would like to show that
$$ Im(m(z)) \sim \sqrt{K + y} $$ for $z = x+ iy$, $K = ||x|-2|$ and $|x| \leq 2$. ( $a \sim b$ means there exist constants $c, C$ such that $cb \leq a \leq Cb$.) Also, $$ Im(m(z)) \sim \frac{y}{\sqrt{K + y}} $$ when $x \geq 2$.
I have tried Taylor expansions with no success.
It is known that $\sqrt{a + ib} = p + iq$ with $p = \frac{1}{\sqrt{2}} \sqrt{\sqrt{a^2 + b^2} + a}$ and $q = \frac{sign(b)}{\sqrt{2}}\sqrt{ \sqrt{a^2 + b^2} - a}$.