Cfrgtkky

I have to calculate two ratios of two pairs of proper integrals, for a range of parameters. I am stuck when it comes to calculating these ratios when the parameters are such that the integrands start to diverge.

Full explanation of the problem follows, but here are my questions:

(1) Can anyone spot a way of finding the ratios of two divergent integrals analytically?

(2) Can anyone help me find a good way of approximating the integrals numerically, and then take the ratio?

(3) I'm not even sure if the singularities are integrable when the integrands start to diverge. How would I test for this?

Any general insights into the problem would be welcome as I'm truly stuck! Full description of the problem below:

I have three integrals to calculate for various values of the parameters $mu$ and $lambda$, over the square $epsilon<x,y<1$ (where the lower limit of the square satisfies $0<epsilonll 1$):

$Omega = int_epsilon^1 dx int_epsilon^1 dy frac{r^{10}}{D(x,y,mu,lambda)}$,

$E = int_epsilon^1 dx int_epsilon^1 dy frac{r^8}{D(x,y,mu,lambda)}$,

$Z = int_epsilon^1 dx int_epsilon^1 dy frac{x^4+5 x^2 y^2}{D(x,y,mu,lambda)}$,

where $r^2=x^2+y^2$ and the integrands each have the denominator $D(x,y,mu,lambda)=mu r^8+r^{10}+lambda(x^2+5x^2y^2)$.

I am really interested in the ratios $F(mu,lambda)=frac{Omega}{E}$ and $G(mu,lambda)=frac{Omega}{Z}$ but, obviously, I must calculate $Omega,E,Z$ for chosen values of $mu,lambda$, and then take ratios.

For physical reasons $D$ cannot be negative. This determines the range of $mu$, $lambda$ I am allowed, in the following way. Let $lambda$ be fixed. Then I can choose any $mu$ as long as
$mu > -r^2 - lambdafrac{x^2+5x^2y^2}{r^{8}}$ for all $x,y$ in the square $epsilon<x,y<1$, i.e. there is a critical value $mu^*(lambda) = mathrm{max}_{x,y} left( -r^2 - lambdafrac{x^2+5x^2y^2}{r^{8}} right)$ which is achieved when the position vector takes the value $(x^*(lambda),y^*(lambda)) = mathrm{argmax}_{x,y} left(-r^2 - lambdafrac{x^2+5x^2y^2}{r^{8}}right)$.

To be clear: $D(x^*,y^*,mu^*,lambda)=0$ so this is where my integrands first become singular. I have implicit expressions for $mu^*, x^*, y^*$ for all values of $lambda$ but I don't think they're important necessarily, suffice it to say that as $lambda$ swings from $+infty$ to $-infty$, the vector $(x^*,y^*)$ tracks down the left-hand edge of the square, i.e. $(epsilon, 1)$ through $(epsilon,y^*(lambda))$ to $(epsilon,epsilon)$.

So the procedure I have is to fix $lambda$ and then for $mu>mu^*(lambda)$ calculate my integrals numerically and take their ratios. The problem is when I approach $mu^*$, when the integrands all become singular at the point $(x^*(lambda),y^*(lambda))$.

Numerical methods (naive ones, at least) break down here, so my questions are: is there anything I can do analytically to find the ratios at $mu^*$? If not can I be more clever about the numerics? And I have no feel for whether the singularities are actually integrable here. Are they?