Evaluate: $$\iint_{\mathbb{R}^2}e^{−(19x^2+2xy+19y^2)}\ dxdy$$
To be honest I am totally stuck and have no idea how to start. Any hint, help would be appreciable.
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2Could you do it if the $xy$ term were absent? – saulspatz Apr 05 '21 at 12:53
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The quadratic form $19x^2+2xy+19y^2$ is positive definite, so the integral converges. Its matrix $$A=\pmatrix{19&1\cr1&19\cr}$$ has determinant $19^2-1^2=360$. So the answer is $\pi/\sqrt{360}$. – Jyrki Lahtonen Apr 05 '21 at 13:08
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@jyrki I don't understand your arguement,it seems to me that I am missing some important information which you know,..can you please elaborate it a more – Ibrahim Islam Apr 05 '21 at 14:51
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Is there any a priori which you used here? – Ibrahim Islam Apr 05 '21 at 14:54
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1See https://math.stackexchange.com/q/384732/321264 and its linked posts. – StubbornAtom Apr 05 '21 at 15:42
1 Answers
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Hint 1: $$ Ax^2+Bxy+Cy^2 = \alpha (x+\beta y)^2 + \gamma y^2, $$
for some $\alpha,\beta,\gamma$.
Hint 2: If $\alpha,\gamma > 0$, you can use linear substitution to get $u^2+v^2$ in the exponent. Can you solve the integral with $u^2+v^2$?
Vasily Mitch
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