Computing the integral $\int e^{2i(a-b)x}e^{-\frac{1}{2}x^2}\mathrm{d}x$

Question

I am pretty stuck when I tried to calculate the Wigner function for the coherent state. Below is part of the equation that I find very challenging.
$$ \int e^{2i(a-b)x}e^{-\frac{1}{2}x^2}\mathrm{d}x\int e^{2i(c-d)y}e^{-\frac{1}{2}y^2}\mathrm{d}y $$ where $a,b,c,d$ is real numbers and $i$ is complex number index.

Please show me a way to solve it.

Possible duplicate of Proof of certain Gaussian integral form — tired, Oct 05 '15 at 09:48
Possible duplicate of $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis — TBBT, Oct 05 '15 at 09:56

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

2

HINT: $$\int e^{2i(a-b)x}e^{-\frac{1}{2}x^2}dx = e^{-2(a-b)^2}\int e^{-\frac{1}{2}(x - 2(a-b)i)^2}dx.$$

And you can follow this question.

edited Apr 13 '17 at 12:21

Community

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answered Oct 05 '15 at 07:02

GAVD

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Computing the integral $\int e^{2i(a-b)x}e^{-\frac{1}{2}x^2}\mathrm{d}x$

1 Answers1