37 



APPENDIX 1 



EVALUATION OF INTEGRALS IN Fj AND M ^ 



REFERENCE FORMULAS 



The associated Legendre functions satisfy certain difference relations which are tabu- 

 lated here for reference (Bateman, Reference 13, p. 360). 



(n-s + l) P„%^ - (2n +l)f,p^+(n+ s) P^_^ = [94] 



yi - t? P^+^ = 2s^Pl-(n+ s)(n- s+ 1) ^1 - ^2 p^-i [95] 



Pn-i =iMP'„ + (n-s+l) VTV ps-i [96] 



P„%1 =piP^ +{n + s)VlT^P^-l [97] 



x/TV /^+^ = U + s + l)(iP^ -{n-s + DP^^j [98] 



(1 - ^2)dPl^ (n + D^p^ - (n - s + 1) P„\i [99] 



a fi 



^ dP^ . [100] 



We shall also need the following integrals: 



r P„"P^c?^ = n^m [101] 



J _l " 2n+l {n-s)\ 



r\p;)2_^.i^^Ll^ [104] 



