In the context of a quantum field theory problem I have a ("nice")* function $\,f: \mathbf{R}^3\rightarrow\mathbf{R}$ that satisfies the following three properties:
- In spherical coordinates it is zero outside of radius $r_0$.
- Its integral over all space is zero: $$\int_{\mathbf{R}^3} f(\mathbf{x}) \,\mathrm{d}^3\mathbf{x} = 0$$
- It satisfies $$\int_{\mathbf{R}^3} |\mathbf{x}|^2f(\mathbf{x}) \,\mathrm{d}^3\mathbf{x} = C \neq 0.$$
I want to integrate this function against (alternating) even powers of $\vert \mathbf{x} \vert$ and $\mathrm{e}^{-i \mathbf{p} \cdot \mathbf{x}}$; i.e. $$\int_{\mathbf{R}^3} \mathrm{d}^3\mathbf{x} \, \mathrm{e}^{-i \mathbf{p} \cdot \mathbf{x}} f(\mathbf{x})(1 - \vert \mathbf{x} \vert^2 + \vert \mathbf{x} \vert^4 - \vert \mathbf{x} \vert^6 + \, ...). $$
But I am paralyzed at this step. I've had the following ideas that have not panned out:
- Somehow cleverly employ Parseval's theorem by recognizing that the above expression is a fourier transform of the equations in properties 2 and 3.
- Somehow cleverly employ integration by parts; unfortunately I don't know how to deal with the expression for the antiderivative of $\,f(\mathbf{x})(1 - \vert \mathbf{x} \vert^2 + \vert \mathbf{x} \vert^4 - \vert \mathbf{x} \vert^6 + \, ...)$ since I only have its integral over all space.
- Write the argument of the exponential as $-i p x\cos{\theta}$ and integrate over spherical coordinates assuming that $f$ is isotropic, i.e. $f(\mathbf{x}) = f(\vert\mathbf{x}\vert).$ This has the obvious drawback of imposing another condition on $ \,f$ as well as putting the integral in a form where I can't use either of the above properties since the integration measure is no longer $\mathrm{d}^3\mathbf{x}$ after I integrate out the $\theta$ coordinate.
Any help would be appreciated; even in the form of informing me there probably isn't enough information to carry out the integral as written.