618 



the circle (rj, to which (?') corresponds for the series P belonging to 7'. 

 As a rule 7\x"'v) will further, together with Tv, be regular in a 

 domain. For let T be ?it>r/>?a/, such that the N.F'.O. is a circle (^f)^(rj), 

 and the N.F.F. a circle (o) <^ (r). If we know that all functions 

 belonging to (o) form part of the F.F., nothing is to be demon- 

 strated as in it is included that both v and .v"^v have a traiismuted 

 in iu). But if it is provisionally only given that v forms part of the 

 F.F. we may explain in the following way that this as a lule 

 includes that .c"'v as well forms part of it ^). If 



ï' = c„ + Cj ?/ 4- . . + cn ;/" + ... {y —X — x,\ 



be the power series in which v may be developed in the vicinity 

 of X ^ .i\, we ha\^e according to the proposition of N". 18, in the 

 domain (a) 



Tv = c, $/ + r-j ^', + . . . -f c„ ê,/ H- . . ., 



if '^'„ be the transmuted of {.i — .rj". If now T is applied term by 

 term to the development of the series of yv, we arrive at 



^"o ^i' H- <^i ^2' + . . . 4- c„ §',i_i + . . . 



From the convergence of the first series that of the second follows 

 if the quotient of $',,+1 by £'„ remains within finite limits for all 

 integral n values. This now is often the case, (for instance with S^, 

 and Z)~^) and if so, it is clear that the series, emanating when 

 y'«z; is substituted for v, converges as well. If, for a moment, v/e 

 write the result as 7\{i/"'v), this rié^^c/ not be equal to 7' (y"'i;) for other 

 values than m =z 0, but in any case a normal transmutation does 

 exist, namely 1\, which, with preservation of the N.F. {a) ^ (/■,), 

 contains both ?; and y^"v, and consequently x'"v in its F.F., and it 

 is evident that T in common cases will be defined in such a way 

 as to be exactly that transmutation. 



If now Tv and T [x^^v) be regular in a domain («) greater than 

 (/•j), this is necessarily the case for a'm =^ 7' '"(v), and there is every 



reason to expect that (/'i= lim \a',„\/'^ will be limited in such a 

 greater circle, consequently that the series P is complete in it. Hence 

 we arrive definitely at the statement : 



The series P^ belonghig to the transmutation 2', ^: {Tv) is not 



^) It is not inconsistent with our agreement made in the beginning of N". 19 

 that we speak of the case in which T{x"^v) is not at the same time regular with 

 To in domains greater than (r^) It only follows from that agreement that we only 

 take notice of a T that for all functions with the radius of convergence r produces 

 a transmuted regular in a domain smaller than (r^), because to the latter a domain 

 smaller than (r) corresponds for the series P. 



