Use Reciprocity Relations to Find Fourier Transform

[erok81]

Homework Statement

Use the reciprocity relations and known transforms to compute the Fourier Transform of the given function.

f(x)=\frac{1}{1+x^{2}}

Homework Equations

With the help of the table of Fourier transforms, write the given functions as F(f).

The Attempt at a Solution

First off my text just starts using that capital F symbol but never really says what it is. I am assuming that is the Fourier Transform?

The hint gives me
\sqrt{\frac{\pi}{2}}e^{-| \omega |}

I am just assuming again, since that is the Fourier transform from the table, it is also F(f)?

If that is right, I can now move onto the reciprocity portion. First...nowhere in my book mentions anything about these. But it does give another hint.

Reciprocity relation: F²(f)(x) = f(-x) where F²(f) = F(F(f)).

Is my F(f) term correct first? If so, then I am not sure what this hint even does for me.

If I had something like f(g(x)) I can do these because I have a known variable to work with and swap in the different functions. But with F(F(f)) I have no idea what that means let alone can solve it.

That's about where I am so far. Not much at all unfortunately. Any ideas or hints to get moving on this problem?

 

[I like Serena]

Homework Statement

Use the reciprocity relations and known transforms to compute the Fourier Transform of the given function.

f(x)=\frac{1}{1+x^{2}}

Homework Equations

With the help of the table of Fourier transforms, write the given functions as F(f).

The Attempt at a Solution

First off my text just starts using that capital F symbol but never really says what it is. I am assuming that is the Fourier Transform?

Yes, it is the Fourier Transform.
Note that the result is again a function, but now from a different variable, usually k, omega, or xi. Apparently in your case omega has been chosen.

The hint gives me
\sqrt{\frac{\pi}{2}}e^{-| \omega |}

I am just assuming again, since that is the Fourier transform from the table, it is also F(f)?

Actually, this is your final answer (with omega as the chosen transformed variable).
You can see this for instance here:

http://www.wolframalpha.com/input/?i=fouriertransform[1/(1+x^2),+x,+\omega]

If that is right, I can now move onto the reciprocity portion. First...nowhere in my book mentions anything about these. But it does give another hint.

Reciprocity relation: F²(f)(x) = f(-x) where F²(f) = F(F(f)).

Is my F(f) term correct first? If so, then I am not sure what this hint even does for me.

Yes, you formula is correct.
To know what this hint does for you, you need to check your table of known transforms.
Are there any entries of the form e^-|x| or e.g. 1/(1+k²)?

If I had something like f(g(x)) I can do these because I have a known variable to work with and swap in the different functions. But with F(F(f)) I have no idea what that means let alone can solve it.

That's about where I am so far. Not much at all unfortunately. Any ideas or hints to get moving on this problem?

If you need more information about Fourier Transforms, you can look e.g. here:

http://en.wikipedia.org/wiki/Fourier_transform
http://mathworld.wolfram.com/FourierTransform.html

Don't you have a textbook that explains a little more about it?

 

[erok81]

Thanks for the reply.

Yes, it is the Fourier Transform.
Note that the result is again a function, but now from a different variable, usually k, omega, or xi. Apparently in your case omega has been chosen.


Actually, this is your final answer (with omega as the chosen transformed variable).
You can see this for instance here:

http://www.wolframalpha.com/input/?i=fouriertransform[1/(1+x^2),+x,+\omega]

I think I was confused because it seemed to be a two part problem. I find the transform in the table then use reciprocity relations to find the transform. In my answer I just pulled the transform from the table and was done.

Yes, you formula is correct.
To know what this hint does for you, you need to check your table of known transforms.
Are there any entries of the form e^-|x| or e.g. 1/(1+k²)?

Yep...there is an entry for e^-|x| which results in √(2/Π) a/(a²+ω², just like your post.

The book gives...

g=F(\frac{\pi}{2}e^{-|x|}), ~F(g)=(\frac{\pi}{2}e^{-|\omega|})

Now that I type out that answer I think I am seeing where the hint comes into play. Except my table gives it in ω so I am already there like you said.

If you need more information about Fourier Transforms, you can look e.g. here:

http://en.wikipedia.org/wiki/Fourier_transform
http://mathworld.wolfram.com/FourierTransform.html

Don't you have a textbook that explains a little more about it?

Thanks for posting this as well. I do have a text but it didn't say much about this problem. The links will help though.

 

[I like Serena]

Yep...there is an entry for e^-|x| which results in √(2/Π) a/(a²+ω², just like your post.

The book gives...

g=F(\frac{\pi}{2}e^{-|x|}), ~F(g)=(\frac{\pi}{2}e^{-|\omega|})

Now that I type out that answer I think I am seeing where the hint comes into play. Except my table gives it in ω so I am already there like you said.

Well, you're not quite there yet.
There's a difference between the Fourier Transform and the Inverse Fourier Transform.
That is, you can not simply swap x and omega in the formulas.

That is why you still need to combine the formula from the table with the reciprocal theorem...