I need some help with this also:
Prove that :
FourierTransform[e^(-ax^2)]=(sqrt[Pi/a]) e^(-ω^2/4a)
Put
$\displaystyle
f(x)=e^{-ax^2}
$
Differentiate:
$\displaystyle
\frac{df}{dx}=-2 a x f(x)
$
Now take FTs of both sides and apply the derivative rules to the forward
and backwards transform:
$\displaystyle
i \omega F(\omega) = -2 a i\frac{dF}{d\omega}
$
so we have $\displaystyle F(\omega)$ satisfies the ODE:
$\displaystyle
\omega F(\omega) = -2 a\frac{dF}{d\omega}
$
and all one now has to do is show that the suggested form for $\displaystyle F(\omega)$ satisfies
this equation (or you could just solve the equation - it's of variables separable type).
(there is at least one other way of doing it, but that requires Cauchy's integral theorem from complex analysis)
RonL