Question

I am sorry to bother the community with such a narrow question, it may perhaps be a little specific. As I study Random Matrix Theory, I often have to solve integrals of the form

$$\mathcal{P} \int_a^b dy \frac{\sqrt{P(y)}}{x-y}$$

Where $P(y)$ is a polynomial positive between $a$ and $b$ and $a\leq x\leq b$. Usually Mathematica does the trick, although it takes an ungodly amount of time for it to compute CPV of an integral. My question is this: does this kind of integral has a name? Does anyone know literature on the subject that might be of use?

For $P(y)=(b-y)(y-a)$ this integral has a simple and elegant value, but for any polynomial larger than degree 2 I can't find any answer. In particular, my ambitions are small and I care more for the case $P(y)=(b-y)(y-a)(c-y)(d-y)$ with $c$ and $d$ outside $[a,b]$. I apologize if it is trivial.

Answer 1

Your principal value integral has the form of a Hilbert transform. It is probably more helpful to note that it is closely related to the Stieltjes transform. The Stieltjes transform of a function $f(y)$ is defined by

$$ S(z)=\int_I dy\frac{f(y)}{z-y} $$

for a real interval $I$ and complex $z$ not in $I$. If we take $I=[a,b]$ and $f=\sqrt{P}$, then your integral is given by

$$ {\cal P}\int_{a}^{b}dy\frac{\sqrt{P(y)}}{x-y}=\lim_{\epsilon\rightarrow 0}\Re S(x+i\epsilon). $$