- #1

Risborg

- 4

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## 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.

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