Does anyone know how to evaluate integrals of them following form:
$$\int_{0}^{\pi} \int_{0}^{2 \pi} (\cos{\phi})^{n_1} (\sin{\phi})^{n_2} (\cos{\theta})^{n_3} (\sin{\theta})^{n_4}Y^m_l Y^p_q Y^i_j \sin{\theta}\; d\phi\; d\theta.$$
Where $Y_l^m$ is the spherical harmonic of degree $l$ and order $m$. The
symbols $n_1$, $n_2$, $n_3$, $n_4$ $l$, $m$, $q$, $p$, $i$, and $j$ are
integers. For any given combination of these integers it is not hard to evaluate the integral, but I need to find all results for a large number of different values of these integers. So many in fact that even tools like Mathematica become to slow.
My current idea is to make some kind of selection criteria for when the integral should be zero, based on symmetry considerations, and then evaluate the non-zero integrals by help of Mathematica. However, if anyone knows any result that would simplify my efforts that would be great.
Best Answer
We are solving the integral:
$$I=\int_{0}^{\pi} \int_{0}^{2 \pi} (\cos{\phi})^{n_1}(\sin{\phi})^{n_2}(\cos{\theta})^{n_3}(\sin{\theta})^{n_4}Y^u_p Y^v_q Y^w_r \sin{\theta}\, d\phi\, d\theta.$$
Due to the nature of the spherical harmonics, which can be written as a product of associated Legendre polynomials and an exponential :
$$Y_l^m(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}\cdot\frac{(l-m)!}{(l+m)!}}P_l^m(\cos\theta)e^{im\phi},$$
you can split the integral in two parts:
$$\Phi(n_1,n_2,u+v+w)=\int_0^{2\pi}(\cos\phi)^{n_1} (\sin\phi)^{n_2} e^{i(u+v+w)\phi}d\phi,$$
$$\Theta(n_3,n_4,p,q,r,u,v,w)=\int^1_{-1} x^{n_3}(1-x^2)^\frac{n_4}{2} P_p^u(x)P_q^v(x)P_r^w(x)dx.$$
The first integral $\Phi$, can be solved by converting the cosine and sine into their exponential form and using the Binomial theorem and taking into account that
$$\int_0^{2\pi} e^{im\phi}d\phi=2\pi\delta_{m,0}.$$
Thus,
$$\Phi(n_1,n_2,m)=\frac{2\pi}{2^{n_1+n_2}i^{n_2}}\sum_{k_1=0}^{n_1}\sum_{k_2=0}^{n_2}(-1)^{n_2-k_2}\binom{n_1}{k_1}\binom{n_2}{k_2}\delta_{n_1+n_2-m,2(k_1+k_2)}.$$
This gives you your first selection rules:
To solve the integral of $\Theta$, you should try to reduce it to the form :
$$\int^1_{-1}P_p^u(x)P_q^v(x)P_r^w(x)dx,$$
which has a closed-form expression. It is related to 3j-symbols/Clebsch-Gordan coefficients and is known as Gaunt's Formula (See Dong S.H., Lemus R., (2002), "The overlap integral of three associated Legendre polynomials", Appl. Math. Lett. 15, 541-546.).
The integral of $\Theta$ can be reduced to this form using the recurrence relations of the associated Legendre polynomials:
$$ P_{l-1}^m(x) - P_{l+1}^m(x) = (2l+1)\sqrt{1-x^2}P_l^{m-1}(x)$$
$$ (2l+1)xP_l^m(x) = (l-m+1)P^m_{l+1}(x)+(l+m)P^m_{l-1}(x)$$