Solving Fourier Transform Problem: f(x) = e^(-pi*x^2)

In summary, the conversation is about solving Fourier transform problems, specifically integrating a function with an exponential term. The first problem involves completing the square and using the result to find the Fourier transform. The second problem involves an additional term and hints are given to solve it.
  • #1
galipop
51
0
Hi All,

I've been going through a few Fourier transform problems and I'm stuck with integrating this one:

f(x) = e^(-pi*x^2)

then

F(e^(-pi*x^2)) = integral (e^(-pi*x^2) * e^(-i*w*x)).dx

Can anyone help me out?

Many Thanks,

Pete
 
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  • #2
1. You should first of all "complete the square" in the exponent.
2. If you've done that, and still got problems about how to evaluate the expression, try to explain what your problem is precisely.
 
  • #3
The Fourier transform of a gaussian is a gaussian, complete the square and do the integral.
 
  • #4
cheers... once I completed the square it was fairly straight forward.
 
  • #5
I have another problem to solve, but it looks similar to the one above.

f(x) = x * e^(-pi*x^2).

So now there is an extra term.

Can I use the result from the previous problem to find the Fourier transform? Any hints to get me started would be greatly appreciated.

Cheers,

Pete
 
  • #6
Note that you easily may replace f with some derivative, dG/dx.
Use the product rule for integration to compute the answer.
 

FAQ: Solving Fourier Transform Problem: f(x) = e^(-pi*x^2)

What is the Fourier Transform of f(x)?

The Fourier Transform of f(x) is a mathematical operation that decomposes a function into its constituent frequencies. In this case, the Fourier Transform of f(x) = e^(-pi*x^2) is a Gaussian function centered at 0 with a width of 1/√π.

How do you solve for the Fourier Transform of f(x)?

To solve for the Fourier Transform of f(x) = e^(-pi*x^2), you can use the formula F(ω) = ∫ f(x)e^(-iωx)dx, where ω represents the frequency variable. In this case, the integral can be evaluated using the Gaussian integral formula and the result will be the Fourier Transform of f(x).

What is the significance of the Fourier Transform of f(x)?

The Fourier Transform of f(x) = e^(-pi*x^2) is important in signal processing and image analysis. It allows us to analyze the frequency components of a signal or image, which can be useful in filtering, noise reduction, and compression.

Can the Fourier Transform of f(x) be used to reconstruct the original function?

Yes, the inverse Fourier Transform, given by f(x) = (1/2π)∫ F(ω)e^(iωx)dω, can be used to reconstruct the original function from its Fourier Transform. However, in this case, since the Fourier Transform of f(x) = e^(-pi*x^2) is a continuous function, the reconstruction may not be exact.

Are there any other methods to solve the Fourier Transform of f(x) = e^(-pi*x^2)?

Yes, there are other methods such as using the Fourier Transform table, using properties of the Fourier Transform, or using software or online calculators. These methods can provide a quicker and more accurate solution compared to manual integration.

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