ABSTRACT

The temporary restriction is removable by analytic continuation: equation (14.12) holds for complex z, provided that QF(z) has its principal value and the branches of (zZ - 1)'i2 and C are continuous in the cut plane. 14.3 An integral for Q;(z) in terms of P,"(z) (with principal branches in each case) can be found from their Wronskian formula.' From (12.16), (13.141, and (14.04) we derive

(n+m)! 1 P," (2) Q~'(z) - Qr (2) P ~ ( z ) = (-)m-- (1 4.13) (n-m)! z2-1

By repeated applications of Rolle's theorem, we see from (14.02) that the zeros of P,"(z) all lie in the interval [-l,l]. Hence on dividing (14.13) throughout by {P:(Z))~ and integrating, we find that

provided that the path does not intersect the cut [- 1, 11. Again, provided that z does not lie on the cut, an integral for Q:(z) involving

P,"(z) in a different way may be found by means of Cauchy's integral formula. For simplicity, restrict m to be zero. Then

where V , is a large circle and V 2 is a closed contour within W1 which itself contains the interval [ - I , ] ] but not the point z ; see Fig. 14.1. The contribution from V1 vanishes as the radius of V , tends to infinity; compare (12.09). Then collapsing '3, onto the two sides of the interval [- 1 , I ] , we find that

By encircling the logarithmic singularity of Qn(t) at t = 1 , we derive from (14.10)

Thus we have Neumann's integral:

14.4 The last result to be established in this section is the so-called addition theorem for Legendre polynomials.