Oru — Solutions of Differential Equations hy Contour Integrals. 117 



Again, each derivative may be found by differentiating under the sign of 

 integration ; thus the initial value of Z"'.f is given by 



2,rtZ)'\r=eJ^//^(A)fa/0(A), (5) 



where 



^^ (A) = (arXP+'-' + ff,._,Af"" + . . . + a,\f)x^ 

 + {ci,Xf-"'-- + . . . + ffjAP,^! 



+ «,A''^,-i- 

 And, in the contour integral in (5), we may subtract from i/'p(A) the product 

 of (j)(A) and any polynomial ; we may thus replace -i/zp by another polynomial, 

 of order r - 1, viz., the remainder, in the usual sense, when >/'p(A) is divided 

 by <^Q^- When 'p lies between 1 and /• - 1 inclusive this remainder is 



- «oA^-'a;„ - («,A''-' + aoAP->i - ... - {ap_,\v-^ + . . .>:e^_i 



+ (ff,.A''-' + «,-.iA'"- + . . ^Xf + («rA''"- + . . .)XpiX + . . . 



4 ar A''x,-i- (6) 



When the distance of the contour from the origin increases indefinitely the 

 integrand tends asymptotically to the value A"^v^f?A, so that the (initial) value 

 of the integral is lizix,^. 



In the symbolic notation 



U^) 



D-X 



t = 



the quotient, on division by ^(A) is 



-B" - A" 



and the remainder is 



t = 



B -\ 



'\p,^{h) - B",p(X) 

 B-\ ' 



t = 



(7) 



(8) 

 (9) 



§ 1'.3. The above solution obtained from the differejitial eqiiMion. 

 I think it of considerable interest that the above solution may be obtained 

 directly from the differential equation,' thus showing that it is the most 

 general solution. 



Writing the equation, with f as the independent variable, in the form 



^{B')o: = 0, (10) 



where B' denotes djdt', multiply across by eH'-i')dt,- and integrate from to t. 



' Compare Cotter : — "A new method of solving Legendre's and Bessel's equations, 

 and others of a similar type," P. R.I. A., xxvii. A, 1911, p. 157- 



- Here, and at corresponding stages throughout, it would be simpler to omit the 

 factor e^' until just before intej;ration with respect to A. 



[13*] 



