# What is the value of this gaussian-like integral?

What is the value of this integral ?

$$$$\int_{-\infty}^ {\infty} \int_{-\infty}^ {\infty} \exp \left( -\frac{A}{2}(x-y)^2 + B (x-y) \right) dx dy$$$$

Note that $$A,B$$ are Real and positive. I can expand it as $$$$\int_{-\infty}^ {\infty} dx \exp(-A/2x^2+ Bx) \int_{-\infty}^ {\infty} dy \exp \left( -\frac{A}{2}y^2 + y(Ax - B) \right)$$$$ Doing the Gaussian integral in $$y$$ gives $$$$\frac{\sqrt{2 \pi}}{\sqrt{iA}} \int_{-\infty}^ {\infty} dx \exp \left(-A/2x^2+Bx + (Ax - B)^2/2A \right) dx$$$$ Cancelling terms in exponent gives $$$$\frac{\sqrt{2 \pi}}{\sqrt{A}} \exp \left(+B^2/2A \right) \int_{-\infty}^ {\infty} dx$$$$ Basically the area under the line in the $$y$$ direction is (excluding constant factors) the inverse of its probability density in the $$x$$ direction. Summed over $$x$$ it becomes infinite.

Postscript: I later determined that the integral over $$y$$ (which enforces a lagrange multiplier constraint) was not required as the system is underdetermined. So the infinity went away. :-)

• If you claim it is nonconvergent, then would that not be the "value" you claim? If you doubt that, why not just edit your claimed solution into your post, so people can critique it? As is, it makes it come off as you expecting us to do all the leg-work, which is not kosher for this site. Commented Aug 3, 2022 at 19:20
• Yes. I guess so. I might add my workings. I thought someone might recognise it.
– Dom
Commented Aug 3, 2022 at 19:24
• @PrincessEev - I added my workings. If the downvote was yours maybe you can zero it.
– Dom
Commented Aug 3, 2022 at 19:30

Try a simple change of variables: $$u=x-y$$, $$v=x+y$$. You will end up with a one dimensional Gaussian integral in $$u$$, and an integral over the entire domain for $$dv$$. The first one is finite and positive, the second is infinite. When you multiply the two you will get $$+\infty$$.
• The way it looks is like an infinite ridge in the $x+y$ direction, with a cross section that is a Gaussian in the $x-y$ direction. Commented Aug 3, 2022 at 19:34