I am reading the appendix of a paper where they take ordinary generating functions $g(x)=\sum_{n\geq0} a_nx^n$ and use the Stieltjes Perron inversion theorem to determine the weight (density) function such that the sequence $a_n$ are the moments of orthogonal polynomials.
The steps are to:
- find $G(z)=\frac{1}{z}g(\frac{1}{z})=\sum_{n\geq0}\frac{a_n}{z^{n+1}}$
- plug in $z=x+iy$, where $x+iy$ is a complex number in standard form
- find the imaginary part of $G(x+iy)$, call it $Im(G(x+iy))$
- determine what is $-\frac{1}{\pi} \lim_{y\rightarrow0^+}Im(G(x+iy))$
The result should be the density function we desire.
The author provides an example, which I will put below and link here (page 36, Appendix).
Let $g(x)=\frac{1-\sqrt{1-4x}}{2x}$, the Catalan generating function.
Then, $G(z)=\frac{1}{z}g(\frac{1}{z})=\frac{1}{2}\left(1-\sqrt{\frac{z-4}{z}}\right)$.
This next step is where I am lost on how the author got this and would love a walkthrough of how to determine the imaginary part of $G(x+iy)$. The author states
$Im(G(x+iy))=-\frac{\sqrt{2}\sqrt{\sqrt{x^2+y^2}\sqrt{x^2-8x+y^2-16}-x^2+4x-y^2}}{4\sqrt{x^2+y^2}}$.
I have been trying to fill in the details of how he got this generating function, but to no avail. I'm assuming there is a simple algebraic trick that I am missing to get $i$ from under the square root. Any help would be greatly appreciated!