I need to solve the following integrals:

$$\int_0^{2\pi}\text{d}\varphi\int_0^\pi\text{d}\theta\: Y_\ell^{*m}(\theta,\varphi) Y_{\ell'}^{m'}(\theta,\varphi)\sin^3(\theta)$$

where $Y_\ell^m$ are the spherical harmonics. Also this integral:

$$\int_0^1 j_\ell(z_{n\ell}x)\:j_{\ell'}(z_{n'\ell'}x)\:x^4 \text{d}x$$

where $j_\ell$ is the $\ell$ spherical Bessel function of the first kind, and $z_{n\ell}$ is the $n$-th root of $j_\ell$.

I'm looking for a source that can indicate a way to calculate these integrals. I cannot find them in the usual online sources (Wikipedia, Mathematica,etc). If someone knows the solution by heart is also welcomed to write it down.

Edit: You can reduce the spherical harmonics integral to $$\int_{-1}^1P_{\ell}^m(x)P_{\ell'}^{m'}(x)(1-x^2)\:\mathrm{d}x $$ where the $P_l^m$ are the associated Legendre polynomials of order $m$. Yet, I don't see any recursion relation that is going to help me with that. As the polynomials are not orthogonal for $m\neq m'$.

Edit2: I've already solved the spherical harmonics part, I only need the Bessel integral

Edit3: The problem with the spherical Bessel integral is that it needs a recursive relation between the $j_\ell(z_{n\ell}x)$ which is not at all trivial. Any thoughts?

  • $\begingroup$ I would ask this type of question in physics stack exchange. During my time at university i've never seen such integrals in math courses, but of course in physic courses for example in quantum mechanics and subsequent coourses. $\endgroup$ – Fakemistake Jan 23 '18 at 10:25
  • $\begingroup$ @Fakemistake Ok, I'll wait, if not I'll migrate the question. $\endgroup$ – Mauricio Jan 23 '18 at 10:30

The spherical Bessel function can be defined from the Bessel function as ([1], see equation 1):

\begin{equation} j_{n}(z)=\sqrt{\dfrac{\pi}{2z}}J_{n+\frac{1}{2}}(z) \end{equation}

I think that from this equation one can establish a recurrence relation using the original Bessel functions. Additionally, we have the following result ([1], see equation 5):

\begin{equation} \int_0^{2\pi}\int_0^\pi\: Y_\ell^{m}(\varphi,\theta) Y_{\ell'}^{*m'}(\varphi,\theta)\sin(\varphi)\text{d}\varphi\text{d}\theta=\delta_{ll}\delta{mm} \end{equation}


[1] Baddour, N. (2010). Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates. JOSA A, 27(10), 2144-2155.

  • $\begingroup$ Yeah the relationship with Bessel functions is very useful. I can do the I integral when the argument is the same. But I need a relationship between the zeros when the argumentes are not the same. $\endgroup$ – Mauricio Feb 5 '18 at 15:33

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