Solve Fourier Transform Problem: f(t)=e^(-t^2/a^2)

[RyanA1084]

Hi all, I had this problem for homework and it stumped me. It's too late to get points for it, but I'd like to know for future reference.

Find the Fourier transform F(w)=integral from -infinity to infinity of f(t)e^(i*w*t)dt

f(t)=e^(-t^2/a^2)

i=sqrt(-1) w=omega=constant a=constant

This looks sort of like a gaussian integral:

integral of e^(-a*x^2)dx=sqrt(pi/a)

but I couldn't see how to do it...

The answer given by the book is sqrt(pi)ae^(-a^2*w^2/4)

Anyone know how to do this??

 

[troutinthemilk]

I think the thing you want to do here is complete the square in the arguments of the exponential. In this expression:
-t^2/a^2 + iwt = -(t-b)^2 +c.
where you can find b and c. When you've done that you can change variables and you will have and expression for the Gaussian that you are familiar with.

 

[M98Ranger]

To solve this problem, you can use the definition of the Fourier transform, which is F(w)=integral from -infinity to infinity of f(t)e^(i*w*t)dt. In this case, f(t)=e^(-t^2/a^2), so we can rewrite the integral as F(w)=integral from -infinity to infinity of e^(-t^2/a^2)e^(i*w*t)dt.

Next, we can use the property of exponential functions that e^(a+b)=e^a*e^b. In this case, a=-t^2/a^2 and b=i*w*t. So we can rewrite the integral as F(w)=integral from -infinity to infinity of e^(-t^2/a^2+i*w*t)dt.

Now, we can use the Gaussian integral formula mentioned in the post, which is integral of e^(-a*x^2)dx=sqrt(pi/a). In our case, a=-1/a^2, so we can rewrite the integral as F(w)=integral from -infinity to infinity of sqrt(pi/a^2)e^(-(-1/a^2)t^2+i*w*t)dt.

Next, we can factor out the constant sqrt(pi/a^2) and use the definition of the Fourier transform again. This will give us F(w)=sqrt(pi/a^2)integral from -infinity to infinity of e^(-(1/a^2)t^2+i*w*t)dt.

Now, we can use the substitution u=t/a, which will give us du=1/a dt. We can also rewrite the exponential term as e^(-(1/a^2)t^2+i*w*t)=e^(-(u^2-2iu*w))=e^(-(u-i*w)^2). This will give us F(w)=sqrt(pi/a^2)integral from -infinity to infinity of e^(-(u-i*w)^2)du.

Using the Gaussian integral formula again, we can rewrite the integral as F(w)=sqrt(pi/a^2)*sqrt(pi)*a*e^(-a^2*w^2/4). Simplifying this, we get F(w)=sqrt(pi)*a*e^(-a^2*w^2/4), which is the same answer given by the book.

In summary, to solve this Fourier transform problem, we used the definition of the Fourier transform, the property of exponential functions,