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what is the relationship between the total power of a function given in spherical coordinates in the Fourier domain:

$E_k=\int_{\mathbb{R}^3}|F(k,\Theta,\Phi)|^2k^2 \sin(\Theta)\,dk\,d\Theta\, d\Phi$

and the total power of its Fourier pair in the ordinary domain:

$E_r=\int_{\mathbb{R}^3}|f(r,\theta,\phi)|^2r^2 \sin(\theta)\,dr\,d\theta \,d\phi?$

Thanks in advance for any answers!

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What is the relation between $F$ and $f$ (with all $2\pi$'s)? – Fabian Apr 27 '11 at 12:37
$f(r, \theta, \phi) = \int_{R^3}F(k, \Theta,\Phi)e^{i\vec{k}\vec{r}}k^2sin(\Theta)dkd\Theta d\Phi$ – Andy Apr 27 '11 at 12:55
up vote 3 down vote accepted

We can plug the definition $$f(r, \theta, \phi) = \int_{\mathbb{R}^3}F(k, \Theta,\Phi)e^{i\mathbf{k}\cdot\mathbf{r}}k^2 \sin(\Theta)dk\,d\Theta\, d\Phi$$ into the expression for $E_r$ and obtain $$\begin{multline} E_r=\int_{\mathbb{R}^3}dr\,d\theta \,d\phi \left[ \int_{\mathbb{R}^3}F(k, \Theta,\Phi)e^{i\mathbf{k}\cdot\mathbf{r}}k^2 \sin(\Theta)dk\,d\Theta\, d\Phi\right]\\ \times \left[\int_{\mathbb{R}^3}F(k', \Theta',\Phi')^* e^{-i\mathbf{k'}\cdot\mathbf{r}}k'^2 \sin(\Theta')dk'\,d\Theta'\, d\Phi' \right]r^2 \sin(\theta) \end{multline}.$$

Exchanging the order of integration and using the completeness relation $$\int_{\mathbb{R}^3} e^{i \mathbf{k} \cdot \mathbf{r}} d^3 r = (2\pi)^3\delta^3(\mathbf{k}) = \frac{\delta(k) \delta(\Theta) \delta(\Phi)}{k^2 \sin(\Theta)}$$ yields $$E_r= (2\pi)^3 \int_{\mathbb{R}^3} |F(k,\Theta,\Phi)|^2 k^2 \sin(\Theta)\,dk\,d\Theta\,d\Phi =(2\pi)^3 E_k.$$

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Thank you Fabian! – Andy Apr 27 '11 at 19:38

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