The Bessel-Legendre Connection

The spherical harmonics of this section constitute the main elements arising from the application of spherical symmetry to the Helmholtz equation. We have already learned in the previous section that the same is true about the cylinder harmonics arising from the application of rotational and translational symmetry applied to that same equation

Recall from page that, upon letting $\theta\to 0$ and $\ell \to \infty$ in such a way that $\theta\ell$ remains finite, the associated Legendre equation

$$\left[\frac{1}{\sin\theta}~\frac{d}{d\theta} \sin\theta\frac{d}{d\theta} + \ell (\ell +1)-\frac{m^2}{\sin^2\theta}\right] P^m_\ell =0$$

becomes

$$\left[\frac{1}{\theta}~\frac{d}{d\theta} \theta \frac{d}{d\theta} +\ell^2 -\frac{m^2}{\theta^2}\right] P^m_\ell =0\,.$$

Consequently,

$$P^m_\ell \to J_m(\ell\theta )\equiv J_m(kr)\,.$$ (5102)

Furthermore, recall the expression for the translated cylinder wave, Eq.(5.34) on page ,

$$H_\nu (k\vert\vec x-\vec x_0\vert )e^{i\nu (\phi -\varphi_0)} = ... ...infty_{m=-\infty} H_{\nu +m}(kr)J_m(kr_0)e^{i(\nu +m)(\varphi - \varphi_0)}\,.$$

Specialize to the case where the wave is rotationally symmetric around the point $\vec x_0\colon (r_0,\varphi_0)$ . Consequently, $\nu=0$ . This, together with the requirement that the wave amplitudes be finite everywhere leads to

$$J_0(k\vert\vec x-\vec x_0\vert )=\sum^\infty_{m=-\infty} J_m(kr)J_m(kr_0) e^{im(\varphi -\varphi_0)}\,.$$

It is evident that, with the help of Equation 5.102, this is the ``small $\theta$ , large $\ell$ '' asymptotic limit of the spherical addition theorem

$$P_\ell (\cos\Theta ) = \sum^\ell_{m=-\ell}\frac{(\ell -m)!}{(\ell... ...} P^m_\ell (\cos\theta )P^m_\ell (\cos\theta_0) e^{im(\varphi -\varphi_0)}\,.$$

These equations illustrate the relational nature of our knowledge: In the limit of large $\ell$ the spherical harmonics becomes indistinguishable from the Bessel hamonics. Learning about one allows us more readily to grasp the other.

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