#### 8.84 Conical functions

8.840 Let us set

$\nu =-\frac{1}{2}+i\lambda ,$

where λ is a real parameter, in the defining differential equation 8.700 1 for associated Legendre functions. We then obtain the differential equation of the so-called conical functions. A conical function is a special case of the associated Legendre function. However, the Legendre functions

${P}_{-\frac{1}{2}+i\lambda}(x),\text{}{Q}_{-\frac{1}{2}+i\lambda}(x)$

have certain peculiarities that make us distinguish them as a special class—the class of conical functions. The most important of these peculiarities is the following

8.841 The functions ...

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