- #1
NanakiXIII
- 391
- 0
I've got an integral of the form
<br /> \int d^3x e^{-i \vec{k} \cdot \vec{x}} \int d^3k e^{i \vec{k} \cdot \vec{x}} f(\vec{k})<br />
and I'm wondering whether or not the following is a valid approach. I want to move the factor
<br /> e^{-i \vec{k} \cdot \vec{x}}<br />
under the k-integral, so I relabel my k so that I end up with
<br /> \int d^3x \int d^3k e^{i (\vec{k}-\vec{k}') \cdot \vec{x}} f(\vec{k}).<br />
One now just needs to remember that the primed k is actually the same as the old k, but we don't want to integrate over it. Now the x-integral turns the exponential factor into a delta-function, so that we get (apart from some factors of 2\pi)
<br /> \int d^3k \delta(\vec{k}-\vec{k}') f(\vec{k}),<br />
which just yields
<br /> f(\vec{k}') = f(\vec{k}).<br />
Now, maybe it's just me, but what I just did sounds too easy to be true, but I'm not sure what might be flawed about my reasoning. Can anyone tell me whether there is anything wrong with it?
<br /> \int d^3x e^{-i \vec{k} \cdot \vec{x}} \int d^3k e^{i \vec{k} \cdot \vec{x}} f(\vec{k})<br />
and I'm wondering whether or not the following is a valid approach. I want to move the factor
<br /> e^{-i \vec{k} \cdot \vec{x}}<br />
under the k-integral, so I relabel my k so that I end up with
<br /> \int d^3x \int d^3k e^{i (\vec{k}-\vec{k}') \cdot \vec{x}} f(\vec{k}).<br />
One now just needs to remember that the primed k is actually the same as the old k, but we don't want to integrate over it. Now the x-integral turns the exponential factor into a delta-function, so that we get (apart from some factors of 2\pi)
<br /> \int d^3k \delta(\vec{k}-\vec{k}') f(\vec{k}),<br />
which just yields
<br /> f(\vec{k}') = f(\vec{k}).<br />
Now, maybe it's just me, but what I just did sounds too easy to be true, but I'm not sure what might be flawed about my reasoning. Can anyone tell me whether there is anything wrong with it?
