I am looking at the Integral $$\int_{a_2}^{a_1} R(x) \sqrt{a_1-x}\sqrt{x-a_2}\sqrt{x-a_3} dx$$ with $$a_1>a_2>a_3$$ and $R(x)$ is a rational function of $x$ with a pole of second order in $x=c$ where $a_2>c>a_3$. I want to find an expansion for $lim (a_2->a_3)$ and for $lim (a_1->a_2)$ [Note, this case is already answered] as well as an expansion for all the $a_i$ being the same. In one case the pole will be a problem and give a logarithmic divergence, this i calculated. But the problem here is that i want to subtract this divergence, and the 'remainder' seems to not converge... I took the standard transformation to get to the Legendre normal form of elliptic integrals and used expansions i found on the internet. But they are very bad in the case where $a_1->a_2$ (this was checked numerically). Does anyone know how to deal with it? Integration by parts doesn't seem to work for me. Also is it even smart to go to standard elliptic forms for some of the limits?
I am glad for any Ansatz you can give, it doesn't need to be a complete answer.
Too long for a comment
The first case $(a_1\to a_2)$ is more straightforward and contains enough information for evaluation. $$I(a_1,a_2,a_3)=\int_{a_2}^{a_1} R(x) \sqrt{a_1-x}\sqrt{x-a_2}\sqrt{x-a_3} \,dx$$ Making the substitution $x=a_1-(a_1-a_2)\,t\quad(\,t\in[0;1]\,)$ $$I=(a_1-a_2)^2\int_0^1 R\big(a_1-(a_1-a_2)t\big)\sqrt{t(1-t)}\sqrt{a_1-a_3-(a_1-a_2)t}\,dt$$ As we do not have singularities on the interval $[0;1]$ (the pole of $R\big(a_1-(a_1-a_2)t\big)$ lays outside this interval), we can write the leading asymptotic term at once: $$I=(a_1-a_2)^2R(a_1)\sqrt{a_1-a_3}\int_0^1\sqrt{t(1-t)}\,dt+o\big((a_1-a_2)^2\big)$$ $$=\frac\pi8(a_1-a_2)^2R(a_1)\sqrt{a_1-a_3}+o\big((a_1-a_2)^2\big);\,\, a_1\to a_2$$ As for the second case ($a_2\to a_3$), I'm afraid no sound conclusion can be made without specific information about $R(x)$. The problem is that leading $a_2\to a_3$, you get two small parameters $(a_2-c)$ and $(a_2-a_3)$, and at some stage you even get a situation when $(a_2-c)\ll (a_2-a_3)$, and $(a_2-c)$ becomes a main small parameter, affecting integral due to the singularity $R(x)$ at $x\to c$. The integral, for example, may diverge.