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Let $f : \mathbb{R}^3\to \mathbb{R}$ be a continuous function of compact support. Its Fourier transform is

$$\mathfrak{F}[f](k)=\int f(x)e^{ikx}dx=\int f(x)\sum_{n=0}^\infty \dfrac{i^n}{n!}k_{a_1}\dots k_{a_n} x^{a_1}\cdots x^{a_n} dx,$$

thus if one defines $$F^{a_1\dots a_n}=\int x^{a_1}\cdots x^{a_n}f(x)dx,$$

one has that the Fourier transform is determined by these parameters

$$\mathfrak{F}[f](k)=\sum_{n=0}^\infty \dfrac{i^n}{n!} k_{a_1}\dots k_{a_n} F^{a_1\dots a_n}.$$

These parameters are called multipole moments of the function.

Now, on electrodynamics we have another idea of multipole moments. It is related to the spherical harmonics. Indeed if we have a function $f : \mathbb{R}^3\to \mathbb{R}$ we expand it as

$$f(r,\theta,\phi)=\sum_{l=0}^\infty \sum_{m=-l}^l C_{lm}(r) Y_{lm}(\theta,\phi),$$

and we call $C_{lm}(r)$ the multipole moments of $f$.

One example would be the well-known formula

$$\dfrac{1}{|\mathbf{x}-\mathbf{x}'|}=4\pi \sum_l \sum_m \dfrac{1}{2l+1}\dfrac{r_<^l}{r_>^{l+1}}Y_{lm}^\ast(\theta',\phi')Y_{lm}(\theta,\phi),$$

which gives the moments

$$C_{lm}(r)=\dfrac{4\pi}{2l+1}\dfrac{r_<^l}{r_>^{l+1}}Y_{lm}^\ast(\theta',\phi').$$

My question is: how are these two definitions of multipole moments related? Are they even the same thing? Because in the way these two definitions are presented they seem to be really different. I want to see the connection in a general case, not just examples of how specific $2^n$-pole moments of either definitions are related.

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The connection lies in the spherical harmonics $Y_{lm}$ on the sphere $S^d$ being the restriction of harmonic homogenous symmetric polynomials in $\mathrm{R}^{d+1}$, see also this answer of mine.

This means that the set of $F^{a_1\dots a_n}$ with $\sum_i a_i = l$ corresponds to the spherical harmonics $Y_{lm}$ with that $l$ - with one caveat because you do not have the harmonicity condition on the polynomials that kills off the trace of their corrresponding tensor. This trace corresponds to the radial components of the function, and the discrepancy is therefore explained by the radial dependency in the first case being inside the $F$, not inside the expansion coefficient.

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  • $\begingroup$ Thanks for the answer ! There are some points I'm missing yet, though: (1) what about the distinction that on the first approach I've presented the moments $F^{a_1\dots a_n}$ recover the Fourier transform, while in the second the moments $C_{lm}(r)$ recover the function itself? (2) The symmetric polynomials you mention corresponding to the spherical harmonics should actually be connected to the $k_{a_1}\cdots k_{a_n}$ part isn't it? I mean, $F^{a_1\dots a_n}$ are the coefficients, so these would correspond to the $C_{lm}$ instead right? $\endgroup$ – user1620696 Apr 21 '18 at 14:47

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