615 



both to the F. F. of the series P corresponding to T. Hence the 

 general theorem of Taylor for the functional calculus has been 

 proved, in the following form : 



If the series P answering to a n o r m a I additive transmutation 

 T be complete in the circular domain («) that forms the N . F. 0. 

 of 7\ 2\w) may be developed in this domain in the functional series 

 of Taylor, in the point iu^=v, if both point of origin v and increase 

 u {w = vu) belong to the circle (i?) corresponding to {a). ^) 



. 20. We may consider both members of (42) as an operation on 

 the function u, and thus read this formula as a development of the 

 transmutation 1\ ^ Tv, applied to the function u in a "series of 

 Mac-Laurin". Since the result we arrived at in the preceding number 

 states that this series (which we shall indicate by P^, since it 

 answers to the transmutation 1\) converges in the domain ('<) for 

 any function belonging to (/3), provided this is also the case with 

 the fixed function r, the series P^ is likewise complete in 

 the domain («), such that the corresponding domain is at most 

 equal to (,:?). This may also be derived directly from the inequality 

 (41), which a foi'tiori holds for j(^r'w|. For it is evident that, instead 

 of the statement of which that inequality is the expression, we may 

 as well say : Corresponding to any arbitrarily small chosen number 

 f there is an integer n, such that 



I «'« !<(/:? « + f)'«, for m>n,, . . . (41a) 



and from it follows, if the second part of the proof in N". 4 

 l^f communication) is consulted, the completeness of the series P^ in 

 («), with a corresponding domain that at most is equal to (|i). 



We may therefore interpret the result, arrived at in N". 18 in 

 this wa}- : If the series P answering to a n o r m a I additive trans- 

 mutation T be complete in the circidar domain (a) forming the 

 N . F. 0. of 7', with corresponding domaiyi Qi), loe have V that the 

 series P^ answering to the transmutation T^^T{v), in lohich v is 

 a given, f.ved function belonging to the circle {[f), is also complete 

 in {it) lüith a corresponding domain that at most is equal to (^), 2" 

 we have in the domain («) 



T, {u) = P, {u) 



for such fu7itions of the F. F. of T, as belong to (|i). 



We further add the following remark: Since the last equality 



^) We have not expressly stated again that u, i\ w, form part of. the F. F. of 

 T, since this is a matter of course now that these functions belong to (^), 

 according to our agreement in the beginning of this number. 



