Question

find the fourier transform of f(x) =e^-a^2 x^2​

Answer 1

Answer:

Caveat: I'm using the normalization f^(ω)=∫∞−∞f(t)e−itωdt.

A cute way to to derive the Fourier transform of f(t)=e−t2 is the following trick: Since

f′(t)=−2te−t2=−2tf(t),

taking the Fourier transfom of both sides will give us

iωf^(ω)=−2if^′(ω).

Solving this differential equation for f^ yields

f^(ω)=Ce−ω2/4

and plugging in ω=0 finally gives

C=f^(0)=∫∞−∞e−t2dt=π−−√.

I.e.

f^(ω)=π−−√e−ω2/4.

Answer 2

find the fourier transform of f(x) =e^-a^2 x^2