928 



expansion is here also essential if x lies in the domain (-^ «J, for 

 tiie convergence of the series is in this case included as a special 

 result in the one just obtained viz. that the aggregate in question 

 is absolutely converging in (^ «,). The coefficient we treat of is 



therefore equal in value to ^i,,» (t>) : k!, so that we ma}'^ write 



S//,m = |él,7H|?2=^- -^^ ' .... (88) 







where the last member shews the meaning of the symbolic second 

 member: in ^i,,» U) we have to replace x by sj, then to expand 

 the expression 5i_„j (§J, as if §, were a number, in its power-series 

 of Mac-Laurin and finally to substitute indices for exponents in the 

 symbolic powers of §,. As already remarked the formula is valid 

 in any domain {n) the radius of which is not grealer than -g- «,. 

 The* series in the last member, however, converges in the whole 

 domain (oj, since the limit in the left-hand member of (87) for 

 a = «, is, according to the very same inequality, less than «j and 

 §i„j(.r) is regular in the closed domain («J; further explanation may, 

 as above in an analogous case, be omitted. The convergence of 

 the series is also uniform in («J, its terms are regular functions of 

 X in that domain, hence its sum represents also a regular function 

 there. This latter must be identical with lii_,„ {x) since this function 

 is also regular in («,) and the two functions agree already in a 

 finite part of («,) viz. (i «J. Thus, finally, the symbolic formula 

 (88) is valid in the same domain as its counterpart (73). 



The generalization of (88) is at once obvious. Evidently we have 



hll,m = [^ll,.kz^ («9) 



valid in the domain («,), because §„ ,„ belongs to the domain («j. 

 In connection with (88) this gives 



^/7/,m = IL^i,4J?3 (90) 



meaning : §i^,„ [x) has to be developed according to powers of x and 

 x^ must be replaced by ^2,i{j^), then the resulting series, which is 

 absolutely and uniformly convergent in («,), represents the function 

 ^\\,m{x), which is regular in the same domain; again expand this 

 function according to powers of x and replace x^ by §3^; {x) ; the 

 resulting series, which is absolutely and uniformly converging in 

 the domain (a,), represents the function b\\\,m{x), regular in that domain. 

 It will be convenient to observe that the interpretation given just now 

 does not at all differ from the one corresponding to the pair of formulae 

 (88) and (89) before they are replaced by (90), so that there is 

 nothing to be proved in doing this. Matters would be different if we 



