Polar Mode Numbers at a Soft Cone

The characteristic equation for polar mode numbers $\nu$ at a cone with soft surface at polar angle $\vartheta=\vartheta _{{0}}$ (see → Sect.  E.21 ) is:

$$P_{\nu}^{m}(\cos\vartheta _{0})\xrightarrow[{\cos\vartheta _{0}\to x_{0}}]{} P_{\nu}^{m}(x_{0})\;\stackrel{!}{=}0\quad with~integer~m=0,\pm 1,\pm 2,{\ldots}$$

(1)

The difficulties with the evaluation of polar mode numbers $\nu$ may be illuminated by the fact that for positive integers m and n the associated Legendre functions are identically zero, $P_{n}^{m}(x)\equiv 0$ for $m>n$, and they are constant for $m=n$. When $m>n$, eq. (1) holds, but $P_{n}^{m}(x)$ then does not represent a mode since it is a trivial solution. Because there are no other solutions $\nu$ for $m>0$ and $\nu<m$, the transition between modes and trivial solutions is steady.

$-\lg\left|{P_{\nu}^{\mu}(\cos\,\vartheta)}\right|$ over $\nu$ and $\mu$ , for $\vartheta=165^{\circ}$ . Equivalences in the plot labels:

? \(nu\to\nu;mu\to\mu;theta\to\vartheta\)...