Computing or upper bounding a complicated integral

Question

I am stuck in trying to compute or more realistically upper bound the following double integral $$ \int_\mathbb{R}\int_\mathbb{R} \frac{\exp(-(x-a)^2)\exp(-(y-a)^2)}{1+\exp(-x^2)+\exp(-y^2)}dxdy $$ as a function of $a$. Of course a trivial (and $a$-independent) bound can be obtained via $$ \frac{1}{1+\exp(-x^2)+\exp(-y^2)}\leq 1 $$ and then solving the resulting double integral closed form, but I was hoping in something better than that. Any suggestion is appreciated!

Answer 1

Denote your integral by $I$. The described estimate gives $I\le \pi$. More accurately, we have $$ \pi - I = \int_{\mathbb{R}}\int_{\mathbb{R}}e^{-(x - a)^2 - (y - a)^2}\frac{e^{-x^2} + e^{-y^2}}{1 + e^{-x^2} + e^{-y^2}}\,dx\,dy. $$ Since $0 < e^{-x^2} + e^{-y^2}\le 2$ we can write $$ \iint_{\Pi}e^{-(x - a)^2 - (y - a)^2}(e^{-x^2} + e^{-y^2})\,dx\,dy\ge \pi - I \ge \iint_{\Pi}e^{-(x - a)^2 - (y - a)^2}\frac{e^{-x^2} + e^{-y^2}}{3}\,dx\,dy. $$ The integrals in the latter inequalities can be calculated explicitly giving $$ \pi - I \approx\exp(-a^2/2) $$ up to some multiplicative constant.

Answer 2

Let $I$ denote OP's integral. Extending @Pavel Gubkin's computation, for any integer $N \geq 1$ we have

\begin{align*} I &= \sum_{k=0}^{N-1} (-1)^n \int_{\mathbb{R}^2} e^{-(x-a)^2-(y-a)^2} (e^{-x^2} + e^{-y^2})^n \, \mathrm{d}x\mathrm{d}y + R_N \\ &= \pi \sum_{n=0}^{N-1} (-1)^n \sum_{\substack{j,k\geq 0 \\j+k = n}} \frac{n!}{j!k!} \frac{e^{-(\frac{j}{j+1}+\frac{k}{k+1})a^2}}{\sqrt{j+1}\sqrt{k+1}} + R_N, \\ \end{align*}

where the remainder $R_N = (-1)^N \int_{\mathbb{R}^2} e^{-(x-a)^2-(y-a)^2} \frac{(e^{-x^2} + e^{-y^2})^N}{1 + e^{-x^2} + e^{-y^2}} \, \mathrm{d}x\mathrm{d}y$ is bounded by

\begin{align*} |R_N| \leq \sum_{\substack{j,k\geq 0 \\j+k = N}} \frac{N!}{j!k!} \frac{e^{-(\frac{j}{j+1}+\frac{k}{k+1})a^2}}{\sqrt{j+1}\sqrt{k+1}} \leq \frac{2^N}{\sqrt{N+1}}e^{-\frac{N}{N+1}a^2}. \end{align*}

For instance, $N = 2$ gives

$$ I = \pi \Bigl(1 - \sqrt{2} e^{-\frac{a^2}{2}} \Bigr) + R_2 \qquad\text{with}\qquad 0 \leq R_2 \leq \frac{4}{\sqrt{3}} e^{-\frac{2}{3}a^2}. $$