Standard solutions: the associated Legendre functions , , , and . and exist for all values of , , and , except possibly and , which are branch points (or poles) of the functions, in general. When is complex , , and are defined by (14.3.6)–(14.3.10) with replaced by : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when , and by continuity elsewhere in the -plane with a cut along the interval ; compare §4.2(i). The principal branches of and are real when , and .
When and , a numerically satisfactory pair of solutions of (14.21.1) in the half-plane is given by and .
Many of the properties stated in preceding sections extend immediately from the -interval to the cut -plane . This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). The generating function expansions (14.7.19) (with replaced by ) and (14.7.22) apply when ; (14.7.21) (with replaced by ) applies when .