# problem on Fourier transforms

• October 31st 2006, 05:19 AM
jmytil
problem on Fourier transforms
I need some help with this also:

Prove that :

FourierTransform[e^(-ax^2)]=(sqrt[Pi/a]) e^(-ω^2/4a)
• October 31st 2006, 10:13 AM
CaptainBlack
Quote:

Originally Posted by jmytil
I need some help with this also:

Prove that :

FourierTransform[e^(-ax^2)]=(sqrt[Pi/a]) e^(-ω^2/4a)

Put

$
f(x)=e^{-ax^2}
$

Differentiate:

$
\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:

$
i \omega F(\omega) = -2 a i\frac{dF}{d\omega}
$

so we have $F(\omega)$ satisfies the ODE:

$
\omega F(\omega) = -2 a\frac{dF}{d\omega}
$

and all one now has to do is show that the suggested form for $F(\omega)$ 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