Is it possible to symbolically solve a 2D integral of the following form $$\int d \vec{r}e^{-\frac{\left| \vec{r} - \vec{r}' \right|^2}{2c}}e^{-i \vec{k} \cdot \vec{r}} $$ where $\vec{r} = (x,y)$ and $\left| \vec{r} - \vec{r}' \right|^2 = (x-x')^2 + (y- y')^2$ ?
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3$\begingroup$ Please include in your question the code you are attempting to use to solve this problem. $\endgroup$– bbgodfreyCommented Feb 7, 2016 at 13:43
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1 Answer
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The solution is a straightforward application of Integrate.
Integrate[Exp[-((x - x0)^2 + (y - y0)^2)/(2 c) - I (kx x + ky y)],
{x, -Infinity, Infinity}, {y, -Infinity, Infinity}, Assumptions -> c > 0]
(* 2 c E^(-(1/2) c (kx^2 + ky^2) - I (kx x0 + ky y0)) π *)
answered Feb 7, 2016 at 13:50