620 



development in the series P^u will in the domain («), tor a function 

 u belonging to (/?), certainly hold^ if a<^i\, but probably also, if 

 ''i <C " <C '■'• We shall see how this is and calculate to that purpose 

 the coefficients a'ra by means of (45). We arrive at 



a'm = (to— .t;)'« I' {<ua (.c)] ^^ ((o— .f)'« S,^ (v). 

 According to (42) the series P^u becomes therefore 



Sr^ {vu)= > m Sr^ (v) (to — .t-)'« . 



.^mi^ mf 







If in it the factor Soi{v) is brought outside the symbol of sum- 

 mation the remaining series is precisely the series Pu, which 

 answers to the transmutation So,{u). Apart from the factor mentioned 

 the series therefore converges in any domain («) <^ (.4), provided u 

 belongs to the domain (^). With the factor S-m (v) added to it the 

 series converges consequently for such a function in any case, if « 

 is smaller than the radius of convergence ?■' of Sr,,{v). Thus our 

 presumption has been affirmed. 



If for instance at (x) = h .i', we have a^ = (^ .i*)'", consequently 

 a («) = ^ a and ^5 = f «, so that 



' 1 3 ' 



But the circle of convergence (r') of Sij^xiv) has a radius 



r' rn: 2r 



SO that the series P, is complete here in a greater domain than (r J. 



22. As another example we take the transmutation Z>' ^ , also in a 

 vicinity of .i- = 0. In N°. 16 we saw that D~^ is a normal trans- 

 mutation such that the numerical field is an arbitrary circle (a), 

 and the functional one consists of the functions belonging to the 

 same circle. The series P belonging to D ^ was further complete 

 in («) with corresponding domain (2«). Thus if we keep with respect 

 to the fixed function v the same notations as before, we have in 

 this case, 



' 1 2 ' 



while the circle of convergence (r') of D~^ (?;) is the same as the 

 one of (v), so that 



r zzz r. 



We expect consequently that the series P^, which belongs to the 

 transmutation D~^{v), will not only be complete in (or) for « <^ ^ r, 

 which is necessary according to the theorem, but also \ï\r<^a<^r. 

 The result confirms this. Formula (45) gives 



