Fourier Transform of a Gaussian Pulse

[cytochrome]

Homework Statement

Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum.

Homework Equations

The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by

f(t) = exp(-t^2/2)

then its Fourier transform is

F(w) = sqrt(2 * pi)exp(-w^2/2)

The Attempt at a Solution

I'm confused by the relevant equations (given by professor) because the exponents are not dimensionless... Nonetheless I just plugged and chugged to get

sqrt(Δt)exp^(-(pi * Δt * w))^2

since this creates a dimensionless exponent. I'm not sure where to include the 2 from the original Gaussian though...

 

[rude man]

Convolve your Gaussian transform with the transform of exp(iwt).

 

[vela]

Homework Statement

Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum.

Homework Equations

The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by

f(t) = exp(-t^2/2)

then its Fourier transform is

F(w) = sqrt(2 * pi)exp(-w^2/2)

The Attempt at a Solution

I'm confused by the relevant equations (given by professor) because the exponents are not dimensionless... Nonetheless I just plugged and chugged to get

sqrt(Δt)exp^(-(pi * Δt * w))^2

since this creates a dimensionless exponent. I'm not sure where to include the 2 from the original Gaussian though...

Can you show your work? It looks like you made some algebra errors along the way.