# integral of spherical harmonics over cube

The complex solid spherical harmonics can be defined as

$$U_n^m(\boldsymbol{r}) = r^n P_n^m(\cos{\theta}) e^{im\phi},$$

where $r,\theta,\phi$ are the usual spherical coordinates of $\boldsymbol{r}=(x,y,z)$. Note that $U_n^m(\boldsymbol{r})$ is a homogeneous polynomial in $x$, $y$, and $z$ of degree $n$, i.e. $U_n^m(\alpha\boldsymbol{r})=\alpha^n U_n^m(\boldsymbol{r})$. I want to work out the integrals

$$I_n^m := \int_{-1}^1 dx \int_{-1}^1 dy \int_{-1}^1 dz \;U_n^m(\boldsymbol{r})$$

but cannot get it (nor did I manage to have maple get it for me). Note that since the $U_n^m(\boldsymbol{r})$ are homogenous, we can reduce this to a surface integral:

$$I_n^m = \frac{1}{n+3} \int \partial \Omega \;U_n^m(\boldsymbol{r})$$

where the intregral is over the surface of the unit cube. Obviously, $I_n^m=0$ for any odd $n$ or $m$, but I need the full story. (I want to get the multipole moments for a uniform density within a cube).

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Out of curiosity, are you looking for (i) closed formulas, (ii) sums, (iii) recursion formulas, or something else? If sums are enough, is it not feasible to express the $U_n^m(r)$ in Cartesian form and integrate over the cube directly? – Andrew D. Hwang Dec 22 '13 at 13:47
@user86418 (i) or (iii) would be okay. And yes, using the Cartesian form of the harmonics is perfectly okay. – Walter Dec 25 '13 at 13:41
There are numerous notational conventions, and working out gritty details runs the risk of producing less-than-useful formulas. A sketch follows, using the physicists' convention for spherical coordinates ($x=r\cos\phi\sin\theta$, $y=r\sin\phi\sin\theta$, $z=r\cos\theta$). – Andrew D. Hwang Dec 25 '13 at 15:24
Using the first two of these recursion relations gives \begin{align*} (n-m)U_n^m&=(2n-1)zU_{n-1}^m-(n+m-1)r^2U_{n-2}^m,\\ 2mzU_n^m&=-(x-iy)U_n^{m+1}-(x+iy)(n+m)(n-m+1)U_n^{m-1}. \end{align*} Writing $U_n^m=\sum_{i,j,k}a_{ijk}x^iy^jz^k$, recursion formulas for the $a_{ijk}$ (and thus for the desired integrals) are straightforward. – Andrew D. Hwang Dec 25 '13 at 15:24
Yes, I know all of them. What I want is recursion (or otherwise) for the $I_n^m$ ... And of course, $U^m_n=\sum_{i,j,k}a^{nm}_{ijk}x^iy^jz^k$, i.e. the coefficients carry 5 indices, not 3. – Walter Dec 26 '13 at 17:00

Here are details of the strategy outlined in the comments for recursively calculating the coefficients of $U_n^m(\mathbf{r})$. This doesn't fully answer the question, but perhaps it will be useful nonetheless.

To fix notation, let $(r, \phi, \theta)$ denote spherical coordinates with the physicists' convention: $$x = r\cos\phi \sin\theta,\quad y = r\sin\phi \sin\theta,\quad z = r\cos\theta.$$ For non-negative integers $n$ and $m$, write $U_n^m(\mathbf{r}) = r^n P_n^m(\cos\theta) e^{im\phi}$ as $$\sum_{i, j, k = 0}^\infty (a_n^m)_{i,j,k} x^i y^j z^k,$$ with the understanding that the coefficient $(a_n^m)_{i,j,k}$ is zero if any index is negative, or is larger than $n$. (Particularly, each sum is finite.)

The first recursion relation for the Legendre polynomials becomes, upon replacing "$x$" by $\cos\theta$, setting $\ell = n + 1$, $$(n - m + 2) P_{n+2}^m(\cos\theta) = (2n + 3) \cos\theta P_{n+1}^m(\cos\theta) - (n + m + 1) P_n^m(\cos\theta).$$ Multiplying through by $r^{n+2} e^{\sqrt{-1}m\phi}$, $$(n - m + 2) U_{n+2}^m(\mathbf{r}) = (2n + 3) z U_{n+1}^m(\mathbf{r}) - (n + m + 1) r^2 U_n^m(\mathbf{r}).$$ Writing each polynomial $U_n^m$ in terms of its coefficients, shifting indices, and equating coefficients of $x^i y^j z^k$ gives the coefficient relation \begin{align*} (a_{n+2}^m)_{i,j,k} &= \frac{2n + 3}{n - m + 2} (a_{n+1}^m)_{i,j,k-1} \\ &\quad- \frac{n + m + 1}{n - m + 2} \bigl((a_n^m)_{i-2,j,k} + (a_n^m)_{i,j-2,k} + (a_n^m)_{i,j,k-2}\bigr). \end{align*} Since $U_0^0(\mathbf{r}) = 1$, we have $(a_0^0)_{0,0,0} = 1$ and $(a_0^0)_{i,j,k} = 0$ for all other $i$, $j$, $k$.

Since $U_1^0(\mathbf{r}) = z$, we have $(a_1^0)_{0,0,1} = 1$ and $(a_1^0)_{i,j,k} = 0$ for all other $i$, $j$, $k$.

The preceding recurrence determines the coefficients $(a_n^0)_{i,j,k}$.

Similarly, $U_0^1(\mathbf{r}) = 0$, so $(a_0^1)_{i,j,k} = 0$ for all $i$, $j$, and $k$; and $U_1^1(\mathbf{r}) = -(x + \sqrt{-1} y)$, so $(a_1^1)_{1,0,0} = -1$, $(a_1^1)_{0,1,0} = -\sqrt{-1}$, while $(a_1^1)_{i,j,k} = 0$ for all other $i$, $j$, $k$.

The preceding recurrence determines the coefficients $(a_n^1)_{i,j,k}$.

The second recursion relation for the Legendre polynomials becomes, upon replacing "$x$" by $\cos\theta$ and $m$ by $m + 1$, and rearranging, $$\sin\theta P_n^{m+2}(\cos\theta) = -2(m + 1) \cos\theta P_n^{m+1}(\cos\theta) - (n + m + 1)(n - m) \sin\theta P_n^m(\cos\theta).$$ Multiplying through by $r^{n+1} e^{\sqrt{-1}(m+1)\phi}$ and using $r\sin\theta e^{\pm \sqrt{-1}\phi} = x \pm \sqrt{-1} y$ gives $$(x - \sqrt{-1} y) U_n^{m+2}(\mathbf{r}) = -2(m + 1) z U_n^{m+1}(\mathbf{r}) - (x + \sqrt{-1} y)(n + m + 1)(n - m) U_n^m(\mathbf{r}).$$ Writing each polynomial $U_n^m$ in terms of its coefficients, shifting indices, and equating coefficients of $x^i y^j z^k$ gives the coefficient relation \begin{align*} &(a_n^{m+2})_{i-1,j,k} - \sqrt{-1} (a_n^{m+2})_{i,j-1,k} \\ &\qquad= -2(m + 1) (a_n^{m+1})_{i,j,k-1} \\ &\qquad\quad- (n + m + 1)(n - m) \bigl((a_n^m)_{i-1,j,k} + \sqrt{-1}(a_n^m)_{i,j-1,k}\bigr). \end{align*}

Since the $(a_n^0)_{i,j,k}$ and $(a_n^1)_{i,j,k}$ are known for all $n \geq 0$, the coefficients $(a_n^m)_{i,j,k}$ may be found recursively, and the integrals $I_n^m$ computed as noted in the comments.

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This is not what I was looking for. I had worked out a similarly tedious way myself (before asking), but was looking for a more compact formula. Thanks for your efforts anyway. – Walter Dec 28 '13 at 19:06

I've solved the problem numerically using a C++ code and found that not only to odd n's vanish, all m's not divisible by 4 vanish - and to my surprise all quadrupole moments vanish. So the first non-zero term past n=0 is n=4 with m=-4,0,+4, then n=6 with m=-4,0,+4, then n=8 with m=-8,-4,0,+4,+8, etc.

Just curious - do you need this to do multipole calculations in a grid code?

EDIT - also, all imaginary parts vanish for all.

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I wanted to know the multipole moments of constant density cube. So knowing which vanish is only part of the story. I also need the value for those which don't. Preferrably not the numerical value, but a closed formula. – Walter Jun 29 '14 at 15:28