Fourier Transform Help ( f(x) = 1 )

[FeDeX_LaTeX]

Hello,

In the past couple of days I have been looking at how to transform a function f(t) into another function F(s) via the Laplace transform, and have practiced performing simple Laplace transformations such at f(t) = sin(at), sin^at, cos(at), e^atf(t) and so on.

I looked on Wikipedia at a formula for the Fourier transform and it's a bit confusing; I know there are several types, here is a formula I found;

$$f(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi ix\xi}dx$$ for every real number ξ.

So if I say that f(x) = 1, then I end up with $$f(\xi) = \int_{-\infty}^{\infty}e^{-2\pi ix \xi}dx$$.

Integrating, we get

$$f(\xi) = -\frac{1}{2\pi ix \xi}e^{-2\pi ix \xi}$$.

However I'm stuck on how to evaluate this now, as I have to find the evaluation of the function at negative infinity and subtract that from the function evaluated at infinity (assuming ξ > 0, but when I do that I end up with an $$e^{\infty}$$ term when evaluating the function at negative infinity. Does this simply mean that it does not converge? Can you show me an example of a function that does converge? I tried using the Fourier transform on sin(x) earlier today and I ended up with $$f(\xi) = 0$$...

Thanks.

 

[DiracRules]

The simplest theory I've studied to treat properly Fourier transformation requires that the function you want to transform belongs to L^1 or L^2, where L^1 and L^2 are particular spaces where $f\in L^1 \Leftrightarrow \int_a^b |f|<+\infty$ and $f\in L^2 \Leftrightarrow \int_a^b |f|^2<+\infty$
http://en.wikipedia.org/wiki/Lp_space"
http://en.wikipedia.org/wiki/Locally_integrable_function"

For example, you can integrate $e^{-a|x|}\,\,\,a>0$ since it belongs to L^1.
Other functions that can be integrated are:
rect(ax)=1 if ax is in [-1/2,1/2], 0 elsewhere
tri(x)=1-|x|

To integrate f=1 and f=trigonometric function such sine and cosine, you need a more wide theory, the theory of distribution (I think it is called this way).

However, if you need, $\mathcal{F}(1)=\delta(x)$.http://en.wikipedia.org/wiki/Dirac_delta_function"

 

[FeDeX_LaTeX]

Hello,

Thanks for the reply. I think I understand what you're trying to say; that the definite integrals from a to b of the functions has to converge (not go to infinity)?

What are rect(x) and tri(x)?

I have lectures on Dirac's Delta function and I will look at these soon.

Thanks for your help.

 

[hunt_mat]

I think that rect(x) is the rectangle function, that it s step function that is 1 for a given integral and zero elsewhere, I am guessing that tri(x) is a triangle function which I would imagine is a straight line from (-1,0) to (0,1) and another line from (0,1) to (1,0).

The Dirac delta function is the answer to your question though.

 

[DiracRules]

I wrote the definitions next to the functions. However, it is as hunt_mat says.

 

[hunt_mat]

Wow, excellent guess on my part!

I actually saw the definitions after I posted.