460 



According to the theorem proved, we have now in the domain («) 



F{u) — I)'^u), 

 for all functions belonging to (2«), since these functions a fortiori 

 belong to {((), and therefore form part of the functional field of Z)-i. 



The series Fu does not converge, in the circle («), for functions 

 that belong to (a), without belonging to (2«)- On account, however of 

 what we stated at the beginning of this number about the contiiiiiitj 

 of f)—^, the relation (28) must hold, for the functions mentioned, in 

 the domain («); this relation here takes the form 



D-^{it) = limP{g,n{x)) (32) 



m = 00 



We shall illustrate this by an example. Let be given ^ <[«<]1, 

 we have then 



u = = 1 + 2.r + . . . 4- (m + 1) am -\- . . ., 



in which 



<p„, (w) = 1 + 2.^; + . . . 4- (m + 1) .^'", 

 is a function that does belong to («) but not to (2«) The trans- 

 formation D-^ gives for this in the whole domain («) 





(33) 



The series P becomes 



1 ^ / A' A" 



(1—^0 , _ .. 



This series diverges in a certain part of the circle («), for example 

 in the point on the real axis d' = a, since « ]> è- We shall show 

 now in a direct way that the limit (32) in the ivhole domain («) 

 exists and is equal to (33). 



We therefore write, somewhat extensively 



y^<f.^,n = .v + 2x' + 3.^" + 4a-* + . . . + (^ 4_ l)^.t;«+i 

 — -- f^'„,= — x' — 3w' _ 6a;* — . . . - (m + 1), .t"«+i 

 + ^/f""^= '^■' + 4^^ + . . . + (^ + 1), .^«+1 



^«+1 („0 

 (m -j- !)• 



