# problem on Fourier transforms

• Oct 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)
• Oct 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

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

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