A particular kind of Cauchy Principal Value integral - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:51:45Z http://mathoverflow.net/feeds/question/112582 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112582/a-particular-kind-of-cauchy-principal-value-integral A particular kind of Cauchy Principal Value integral Ricardo Marino 2012-11-16T14:58:17Z 2012-11-16T19:18:29Z <p>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</p> <p>$$\mathcal{P} \int_a^b dy \frac{\sqrt{P(y)}}{x-y}$$</p> <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/112582/a-particular-kind-of-cauchy-principal-value-integral/112606#112606 Answer by Carlo Beenakker for A particular kind of Cauchy Principal Value integral Carlo Beenakker 2012-11-16T18:50:40Z 2012-11-16T19:18:29Z <p>Your principal value integral has the form of a <A HREF="http://en.wikipedia.org/wiki/Hilbert_transform" rel="nofollow">Hilbert transform</A>. It is probably more helpful to note that it is closely related to the <A HREF="http://en.wikipedia.org/wiki/Stieltjes_transformation" rel="nofollow">Stieltjes transform</A>. The Stieltjes transform of a function $f(y)$ is defined by</p> <p>$$S(z)=\int_I dy\frac{f(y)}{z-y}$$</p> <p>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</p> <p>$${\cal P}\int_{a}^{b}dy\frac{\sqrt{P(y)}}{x-y}=\lim_{\epsilon\rightarrow 0}\Re S(x+i\epsilon).$$</p>