I need some help with this also:

Prove that :

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

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- Oct 31st 2006, 05:19 AMjmytilproblem 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 AMCaptainBlack
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