610 Mr. Savidge on the Integration of a Class of 



tJ» + j(a + J ;#) = 



i^S+l[- i ( 1 . + T) U - (ai ' ) -^'-» (fl -* J * ) ] • (16) 



U w (« ; .r) = 

 ^^["(2a-l-*)Ui(a-l;'*) + (n-a+l)IJ.(a-2;^] .... (17) 



U B+ i(a ; #) = 



2/2 + 1 r TT , . r2n(2a+l) 2n(2n-I)T TT , .T 



(18) 



A comparison of the series (3) and (5) shows that 



P m (-a-l;*) = (-l)^U»(a;-a?). . .(19) 



Solution in terms of Bessel Functions*. 

 Ifa=H 



_ g r n + ^ or (tt + i)(rc+i) £ 2 1 



Wl " U{2n) L 2n+l ' |1 (2n + l) (2n + 2) ' |2_ T "" J 



[This may be verified by putting in equation (1) 

 m=l, r=l, 9 =0, P=i] 



A second solution is given by substituting the Bessel 

 function of the second kind, K„, for I„. Thus we have a 

 complete solution in terms of these functions when a=— -J; 

 and it may be obtained whenever 2a is an integer (for any 

 value of n) by means of the relations (7) to (19). 



Solution when argument very great. 



When 30 is very great the series solutions given become 

 laborious to calculate, and we may use the following asym- 

 ptotic semi-divergent expansions in descending powers of the 

 argument. (It will be noticed, from an examination of 



* Tables of I»(#) are given in Gray & Matthews's ' Bessel .Functions.' 

 Kn(x) has been tabulated by Isherwood : Manchester Memoirs (1904),. 

 xlviii. p. 19. 



