Reversion of Power Series. 367 



These operations give 



1 _ / /fAo . 2A, 



+ (n + l)A B +(n + 2)A, l+ tf + ...].. . (3) 



It may be shown by substitution of y from (1) that 

 nA n -iy- l y' is the only term in the right-hand member which 

 contains as' 1 . That this is true is shown also by means of 

 the equation 



f -(tt-i)<fc\y'-V 



— (n— I) da; L v J \ a a J ± 



d.clx 71 - 1 nc n - 2 x J 



a series which after differentiation contains no terms in .r _1 

 for integral values of ft other than unity. Hence, by equating 

 the coefficients of x~ l in (3), we obtain * 



A, 



-m. » 



where the expression in the brackets means the coefficient 

 of x~ l in the development of y~ n as a function of x. This 

 coefficient may be found without much difficulty. Performing 

 the indicated expansion 



y-» = (a Q x + a 1 x 2 + a 2 x 3 + . . .)"* 



=(«„*) -'(i-V-'v 2 + •••)"' 



= (a^)'" [l + »(M + is** +...)+••• 



•f " ( " + 1) • •;, ( " + r ~ 1) (V + ^+ ■ ■ .)'+ ■ ■ .].(5) 



* Harkness and Morley, 'Introduction to Analytic Functions,' 

 p. 344. 



