196 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



3-5 ---- (2fl-l) 



2. 



2-4-6.... 21, 



(Hargreaves, Mess, of Math. 26, p. 89, 1897) 



9.070 The differential equation: 



f 9N d?y dy 



* 



If m is a positive integer, and -i>x> + i, two solutions of this differential 

 equation are the associated Legendre functions 



9.071 If w, w, r are positive integers, and n>m, r>m: 



2 (n + w) ! . 



9.100 Bessel's Differential Equation: 



9.101 One solution is: 



k=o 



9.102 A second independent solution when v is not an integer is: 



y = J- v (x). 



9.103 If v = n, an integer: 



J-n(x) = (-l)Vn(*). 



9.104 A second independent solution when ^ = n, an integer, is: 



(see 6.61). 



