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Evaluate $$ \int^{+\infty}_{-\infty}\, \int^{+\infty}_{-\infty}\, e^{-(3x^{2}+2\sqrt{2}xy+3y^2)}dxdy\,. $$

Sayantan
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    You are meant to share in your post your thoughts on the problem. What have you tried? – StubbornAtom Jan 17 '18 at 14:50
  • Yeah I know Gaussian Integral, that double integral of e^(-x^2-y^2) is pi. But I tried to adjust, but due to lack of time I could not adjust it. Also I thought to factorize the function into to functions of x and y.But I did not get it. – Sayantan Jan 17 '18 at 15:03
  • Plenty of similar problems are posted here with different quadratic forms in the exponent of $e$ in the integrand. Some of them are:https://math.stackexchange.com/questions/2214058/compute-int-infty-infty-int-infty-inftye-x2x-y2y2dxd, https://math.stackexchange.com/questions/2058201/evaluate-i-int-infty-infty-int-infty-infty-mathrm-e-2x22xy2, https://math.stackexchange.com/questions/1096793/int-infty-infty-int-infty-infty-e-left2x22xy2y2-right,https://math.stackexchange.com/questions/877711/evaluate-int-infty-infty-int-infty-inftye-frac12x2-xyy. – StubbornAtom Jan 17 '18 at 15:31
  • @StubbornAtom Thanks a lot. I have not seen it yet. – Sayantan Jan 17 '18 at 15:51
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    Possible duplicate of Compute $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+(x-y)^2+y^2)}dxdy$ – G Cab Jan 17 '18 at 18:38
  • The exercise boils down to the computation of a determinant, see https://math.stackexchange.com/a/2123314/44121 – Jack D'Aurizio Jan 17 '18 at 19:58
  • @JackD'Aurizio Thanks a lot.Actually I didn't know that formula. Yesterday when I checked similar questions, which I didn't come across before, I got the answer. – Sayantan Jan 18 '18 at 04:46

1 Answers1

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HINT:

$$\int_{-\infty}^\infty\exp\left(-\text{n}\cdot x^2-\text{m}\cdot x\right)\space\text{d}x=\frac{\sqrt{\pi}}{\sqrt{\text{n}}}\cdot\exp\left(\frac{\text{m}^2}{4\cdot\text{n}}\right)\tag1$$

When $\Re\left(\text{n}\right)>0$

Now, write your integral in the following form:

$$\int^{+\infty}_{-\infty}\, \int^{+\infty}_{-\infty}\, e^{-(3x^{2}+2\sqrt{2}xy+3y^2)}dxdy=\int^{+\infty}_{-\infty}e^{-3y^2}\, \int^{+\infty}_{-\infty}\, e^{-3x^{2}-2\sqrt{2}xy}dxdy\tag2$$

Jan Eerland
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