In section 7.5,${\mathbf{e}}^{{\mathbf{im}}_{\mathbf{l}}\mathbf{\phi }}$is presented a sour preferred solution to the azimuthal equation, but there is more general one that need not violate the smoothness condition, and that in fact covers not only complex exponentials but also suitable redelinitions of multiplicative constants, sine, and cosine,

$\mathit{\Phi }{\mathit{m}}_{\mathbf{1}}\left(\Phi \right)\mathbf{=}\mathit{A}{\mathbf{e}}^{\mathbf{+}{\mathbf{im}}_{\mathbf{l}}\mathbf{\phi }}\mathbf{+}{\mathbf{Be}}^{\mathbf{+}{\mathbf{im}}_{\mathbf{l}}\mathbf{\phi }}$

(a) Show that the complex square of this function is not, in general, independent of $\mathbf{\phi }$.

(b) What conditions must be met by A and/or B for the probability density to be rotationally symmetric – that is, independent of $\mathbf{\phi }$ ?

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