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 Post subject: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 11:43:40 UTC
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Joined: Sun, 14 Mar 2010 14:12:02 UTC
Posts: 35
Hi,
I am working on the reproduction of a calculation I found in a paper and I am stuck halfway. I have a function:

Where is the fourier transform of i.e.

Now I want to calculate which is defined as

Solving the part between square brackets first I end up with:

Which then results in:

However, in the paper the solution reads:

My question is twofold: (i) is there any mistake in my derivation? (ii) is my answer somehow equivalent to the answer of the paper and, if so, why?

Thanks

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 Post subject: Re: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 13:18:07 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6066
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
michielm wrote:
Hi,
I am working on the reproduction of a calculation I found in a paper and I am stuck halfway. I have a function:

Where is the fourier transform of i.e.

Now I want to calculate which is defined as

Solving the part between square brackets first I end up with:

Which then results in:

However, in the paper the solution reads:

My question is twofold: (i) is there any mistake in my derivation? (ii) is my answer somehow equivalent to the answer of the paper and, if so, why?

Thanks

What answer of the paper?

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 Post subject: Re: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 13:46:51 UTC
 Member

Joined: Sun, 14 Mar 2010 14:12:02 UTC
Posts: 35
My result:

However, in the paper the solution reads:

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 Post subject: Re: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 14:22:44 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6066
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
michielm wrote:
My result:

However, in the paper the solution reads:

I'm not too sure how you get your answer, but the paper's solution looks like a case of missing some modulus signs.

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 Post subject: Re: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 16:03:27 UTC
 Member

Joined: Sun, 14 Mar 2010 14:12:02 UTC
Posts: 35
I get my answer in the following way:
is defined as

Let's first calculate the partial derivates:

and

If I square both I get:

and

The part between square brackets then becomes:

(the first and the last term in cancel against and respectively)

Plugging this into I end up with:

Which then results in:

I hope this is more clear. Is there any mistake in here (or in my thinking)?

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 Post subject: Re: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 16:44:30 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6066
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
michielm wrote:
I get my answer in the following way:
is defined as

Let's first calculate the partial derivates:

and

If I square both I get:

and

The part between square brackets then becomes:

(the first and the last term in cancel against and respectively)

Plugging this into I end up with:

Which then results in:

I hope this is more clear. Is there any mistake in here (or in my thinking)?

Why does
?
You only know it is . Ditto for . But they are inconsequential.

What I am worry about is the y-integral, because it doesn't necessarily converge. Indeed, it does not converge if we are working with . Maybe you are working with ? But if that is the case, what about x? If you do not work on the whole of then you need something extras in your Fourier inversion formula for .

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 Post subject: Re: solving a complex fourier transformed equationPosted: Thu, 17 Nov 2011 17:37:30 UTC
 Member

Joined: Sun, 14 Mar 2010 14:12:02 UTC
Posts: 35
That is a very good point. Can't believe I missed that! Thanks!

About the convergence of the y-integral, indeed the boundaries from to are only formal at the moment. The physics of the problem allow me to put a lower and upper boundary to avoid singularities.

I will see if I can get to the paper's solution now.

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