Integration test of dirac delta function as a Fourier integral

In summary, the problem involves finding the Fourier transform of the Dirac delta function, transforming it back to real space and plotting the result, and testing that the delta function represented by a Fourier integral integrates to 1. The solution involves taking a limit using the Dirichlet Kernel.
  • #1
Risborg
4
0

Homework Statement


Problem:
a) Find the Fourier transform of the Dirac delta function: δ(x)
b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves).
c) test by integration, that the delta function represented by a Fourier integral integrates to 1

Homework Equations


So far I've done a) and b) and the delta function turns out to be [itex] \delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega [/itex]

I've plotted this and it seems to be correct, and I also asked some other students in class and they got the same result, so i don't think that's the issue.

The Attempt at a Solution


So to solve c) I try to integrate δ(x) from -∞ to ∞, but it shouldn't really matter as long as 0 is between the integration limits.
[tex]
\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega x}d\omega dx\\
\frac{1}{2\pi} \int_{-\infty}^{\infty}\left[\frac{1}{ix}e^{i\omega x}\right]_{\infty}^{\infty} dx\\
\frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\frac{1}{2i}(e^{i\infty x} - e^{-i\infty x}) dx\\
\frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\sin(\infty x) dx
[/tex]
Apparently I end up with an integral that's impossible to solve (without approximation), and the sine function has infinity as its argument... So I was hoping you would know where I went wrong.
 
Last edited:
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  • #2
Risborg said:

Homework Statement


Problem:
a) Find the Fourier transform of the Dirac delta function: δ(x)
b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves).
c) test by integration, that the delta function represented by a Fourier integral integrates to 1

Homework Equations


So far I've done a) and b) and the delta function turns out to be [itex] \delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega [/itex]

I've plotted this and it seems to be correct, and I also asked some other students in class and they got the same result, so i don't think that's the issue.

The Attempt at a Solution


So to solve c) I try to integrate δ(x) from -∞ to ∞, but it shouldn't really matter as long as 0 is between the integration limits.
[tex]
\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega x}d\omega dx\\
\frac{1}{2\pi} \int_{-\infty}^{\infty}\left[\frac{1}{ix}e^{i\omega x}\right]_{\infty}^{\infty} dx\\
\frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\frac{1}{2i}(e^{i\infty x} - e^{-i\infty x}) dx\\
\frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\sin(\infty x) dx
[/tex]
Apparently I end up with an integral that's impossible to solve (without approximation), and the sine function has infinity as its argument... So I was hoping you would know where I went wrong.

The last three lines are nonsense; you need to take a limit. Use
[tex]\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_N^N e^{i\omega x}d\omega dx[/tex]
for finite ##N > 0##. Work it through, then take the limit as ##N \to \infty##.

Hint: Dirichlet Kernel.
 

FAQ: Integration test of dirac delta function as a Fourier integral

1. What is the purpose of performing an integration test on the Dirac delta function?

The purpose of an integration test on the Dirac delta function is to verify its properties as a Fourier integral. This test helps to ensure that the function satisfies the conditions required for it to be considered a distribution, and that it can be used to represent a wide range of physical phenomena.

2. How is the integration test of the Dirac delta function performed?

The integration test involves taking the Fourier transform of the delta function and evaluating it at different values of the frequency variable. This is typically done using mathematical techniques such as contour integration or the convolution theorem.

3. What are the main properties of the Dirac delta function that are tested during the integration test?

The main properties that are tested during the integration test include the sifting property, which states that the integral of the delta function over any interval that contains the origin is equal to 1, and the scaling property, which allows the delta function to be scaled by a constant factor without changing its sifting property.

4. What are some potential challenges when performing an integration test on the Dirac delta function?

One potential challenge is dealing with the singularity of the delta function at the origin, which can lead to difficulties in evaluating the Fourier transform. Another challenge may arise when dealing with functions that are not well-behaved, such as those with infinite discontinuities or non-differentiable points.

5. How is the integration test of the Dirac delta function used in practical applications?

The integration test of the Dirac delta function has a wide range of applications in fields such as physics, engineering, and signal processing. It is commonly used to represent point sources, impulsive forces, and other phenomena that can be modeled as impulses. It is also a fundamental tool in the theory of distributions, which has many practical applications in mathematics and physics.

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