Auxilliary Eigenfunctions

Let

$$X = -\frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Phi}_r}{\partial \phi},$$ (12.249)

$$Y = \cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta},$$ (12.250)

$$P ={\mit\Phi}_r',$$ (12.251)

$$Q = {\mit\Psi}_r''$$ (12.252)

in Equations (12.191)-(12.196). It follows that

$$D\,{\mit\Phi}_r' = \frac{1}{\sin\theta}\left[ \frac{\partial}{\partial\theta}\left... ...hi}\left(\cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta}\right)\right],$$ (12.253)

$$D\,{\mit\Psi}_r''&= \frac{1}{\sin\theta}\left[ \frac{\partial}{\p... ...hi}\left( \cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial \phi}\right)\right],$$ (12.254)

and

$$\frac{1}{\sin\theta}\,\frac{\partial {\mit\Phi}_r'(\theta,\phi_\pm)}{\partial\phi} = -\cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta},$$ (12.255)

$${\mit\Psi}_r''(\theta,\phi_\pm) = 0,$$ (12.256)

as well as

$$-\frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Phi}_r}{\partial\phi} = - \frac{\partial{\mit\Phi}_r'}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}_r''}{\partial\phi},$$ (12.257)

$$\cos\theta\,\frac{\partial{\mit\Phi}_r}{\partial\theta} = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}_r'}{\partial\phi} + \frac{\partial{\mit\Psi}_r''}{\partial\theta}.$$ (12.258)

Let

$$X = \cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial \theta},$$ (12.259)

$$Y = \frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Psi}_r}{\partial\phi},$$ (12.260)

$$P ={\mit\Phi}_r'',$$ (12.261)

$$Q = {\mit\Psi}_r'$$ (12.262)

in Equations (12.191)-(12.196). It follows that

$$D\,{\mit\Psi}_r''&= - \frac{1}{\sin\theta}\left[\frac{\partial}{\... ...hi}\left( \cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial \phi}\right)\right],$$ (12.263)

$$D\,{\mit\Psi}_r' = \frac{1}{\sin\theta}\left[ \frac{\partial}{\partial\theta}\left... ...hi}\left(\cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial\theta}\right)\right],$$ (12.264)

and

$$\frac{1}{\sin\theta}\,\frac{\partial {\mit\Phi}_r''(\theta,\phi_\pm)}{\partial\phi} =- \frac{\cos\theta}{\sin\theta}\,\frac{\partial {\mit\Psi}_r(\theta,\phi_\pm)}{\partial\phi},$$ (12.265)

$${\mit\Psi}_r'(\theta,\phi_\pm) = 0,$$ (12.266)

as well as

$$\cos\theta\,\frac{\partial{\mit\Psi}_r}{\partial\theta} = - \frac{\partial{\mit\Phi}_r''}{\partial\theta} -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Psi}_r'}{\partial\phi},$$ (12.267)

$$\frac{\cos\theta}{\sin\theta}\,\frac{\partial{\mit\Psi}_r}{\partial\phi} = -\frac{1}{\sin\theta}\,\frac{\partial{\mit\Phi}_r''}{\partial\phi} + \frac{\partial{\mit\Psi}_r'}{\partial\theta}.$$ (12.268)