463 



br^ (w(^l)) = y m -^ — 



mW(^,), 



if the circle with x^ as centre and 



as radius lies within the domain of convergence of u (c). But this 

 is evidently exactly the case when u belongs to the circle with 

 as centre, and 



^ — a -\- I (O {0C,n ) — Xm \ 



as radius, owing to the signification of x^- We arrive thus at the 

 same result as higher up, so that the theorem of Tayi.or for the 

 functional calculus, in case the operation is a substitution, may at 

 once be deduced from that same theorem of the theory of functions. 

 Now that this has become evident, the fact that the theorem of 

 Taylor is exactly valid for the functional calculus, if the series P 

 in question represents a complete transmutation, appears in a clearer 

 light. For with regard to the convergence of a series, the behaviour 

 of the terms at infinity only is decisive, and we can say now that 

 any series of the nature in question behaves, as regards its terms 

 at infinity, in a certain sense as a substitution. For, since 



1 

 üx =: lim j a,n |"« ' 



we have for large ni approximately (at least for an infinite number 

 of terms) 



SO that, in a definite point x, the modulus of a,n behaves for great 

 values of m as the m^^ power of a positive number aj^. This, now, 

 is also the case for the substitution, where this number is ax = \oi[.r)—x\; 

 only the behaviour in question has already been present here from 

 the first values of m. 



The domain in which the series P corresponding to a transmuta- 

 tion converges for a certain function u is naturally smaller than the 

 circle of convergence (r) of u, because the series P consists of powers 

 of Du. If therefore the numerical field of operation («) of T is 

 greater than the numerical field of the functions {o), Pu will not 

 exist every whei-e in («) for such functions of F{T) as do belong 

 to (ö), but not to («)• But the limit-relation (28), 



Tu — Urn P{ff„,{x)) (28) 



must necessarily be fulfilled everywhere in («) for the last mentioned 

 functions. 



