ABSTRACT
Many of the properties of Pn(z) given in Chapter 2 are capable of extension to the functions P:(z) and Q;(z). We begin with generalizations of Schlafli's integral. 13.2 Theorem 13.1 When z does not lie in the interval (- m, - 11 the principal value of Pv-' (z) is given by
The integration path is depicted in Fig. 13.1. We first observe that it suffices to prove either (13.01) or (13.02); the other repre-
sentation follows immediately from the identity PZt-,(z) = P;"(z). The differential equation satisfied b y w = (z2 - 1)-"I2P;"(z) is found to be
Let us substitute for w by means of a contour integral of the form
We have
Thus I(z) satisfies (13.03) when the content of the square brackets has the same value at the two ends of 8. This condition is fulfilled by the loop integral on the right of (13.01), since the integral converges at the extremities of the path when Rep > Rev, and the content of the square brackets vanishes there. Accordingly, the right-hand side of (13.01) is a solution of the associated Legendre equation.
