# An integral involving many exponential terms with quadratic exponents in the denominator

Given $$k$$ points $$\{p_1,\cdots, p_k\}$$ in $$\mathbb{R}^n$$ and positive constants $$r_1, ..., r_k$$ and another positive constant $$\alpha>0$$. Is there a way to compute/approximate the following (Riemann) integral in terms of $$\{p_1, ..., p_k\}$$, $$\{r_1, ..., r_k\}$$ and $$\alpha$$? $$\int_{\mathbb{R}^n} \frac{dx}{1+\alpha(e^{r_1\lVert x - p_1\rVert^2}+\cdots +e^{r_k\lVert x-p_k\rVert^2})}$$ where $$\lVert v\rVert$$ denotes the standard $$2$$-norm in $$\mathbb{R}^n$$. For example, in $$\mathbb{R}^2$$, I am considering integrals like $$\int_{\mathbb{R}^2}\frac{dxdy}{1+2(e^{2\lVert (x,y)-(3,4)\rVert^2}+e^{5\lVert(x,y)-(6,7)\rVert^2})}.$$ By "approximating the integral", I mean a sequence of expressions $$\{f_i(p_1, ..., p_k, r_1, ..., r_k, \alpha)\}_{i = 1}^\infty$$ in terms of $$\{p_1, ..., p_k\}$$, $$\{r_1, ..., r_k\}$$ and $$\alpha$$, such that $$f_i$$ converges to the value of this integral as $$i\to\infty$$. Any idea or useful reference to related topics is welcome. Thanks

• Do you really want to use $k$ points and let $k\rightarrow \infty$ here, or did you inadvertently reuse "$k$" for two separate quantities? – Michael Engelhardt Oct 16 at 3:06
• @MichaelEngelhardt Confusion fixed. Thanks for pointing out. I made a mistake by using the same notation $k$. – Min Wu Oct 16 at 3:48
• I believe (though do not know how to prove) there is no analytical way to evaluate this integral. Does anyone have any idea for how to approximate it? – Min Wu Oct 18 at 3:05
• Well ... probably this is not the answer you want to hear, but in principle, any valid numerical method implemented on a computer to calculate your integral generates just such a sequence as you describe. Of course, the sequence probably can't be written down easily, and will be only for one fixed set of $p$, $r$ and $\alpha$ each time you use it ... – Michael Engelhardt Oct 18 at 3:17