Mastering Fourier Transform: Solving a Tricky Integral with Expert Tips

[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. I posted in the homework help forum but figured I'd try here too.

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??

 

[arildno]

Complete the square..

 

[M98Ranger]

Hi there,

First of all, don't feel discouraged if you struggled with this problem. The Fourier transform can be a tricky concept to grasp, especially when it involves integrals. But with some expert tips, you'll be able to master it in no time.

To solve this integral, we can use the following property of the Fourier transform:

F(f(t)) = 1/√(2π) ∫f(t)e^(-iwt)dt

Substituting in the given values, we get:

F(f(t)) = 1/√(2π) ∫e^(-t^2/a^2)e^(-iwt)dt

Now, we can use the Gaussian integral you mentioned and substitute it in for the e^(-t^2/a^2) term. This gives us:

F(f(t)) = 1/√(2π) ∫sqrt(pi/a)e^(-(t^2/a^2 + iwt))dt

Next, we can use the property of exponential functions that e^(ab) = (e^a)^b. This gives us:

F(f(t)) = 1/√(2π) ∫sqrt(pi/a)e^(-(t/a + iw)^2)dt

Now, we can use the substitution u = t/a + iw and du = dt/a. This gives us:

F(f(t)) = 1/√(2π) ∫sqrt(pi/a)e^(-u^2)du

And finally, we can use the Gaussian integral again to solve this integral. This gives us the final result:

F(f(t)) = sqrt(pi)ae^(-a^2*w^2/4)

I hope this helps and gives you a better understanding of how to approach Fourier transforms. Remember to always break down the problem into smaller, more manageable steps and use properties and substitutions to your advantage. Keep practicing and you'll become a master in no time!