1101 
(«). In order to. make the difference clearer we give a few examples 
which, at the same time, are proper to dissolve another misunder- 
standing that may have arisen after the reading of the consideration 
in N°. 5 on exceptional cases. These exceptional cases consist in the 
fact that the series (1) for some functions not belonging to (8) pro- 
duces a transmuted in the whole domain (a), or, as we may say 
as well, that that series produces for some functions, which have 
exactly (8) as their circle of convergence, a transmuted in a domain 
greater than («). It occurs however frequently that the transmuted 
of a function w with a radius of convergence 8 exists anywhere 
within a certain circle greater than («), without being determined 
there by a series of the form (1). Y 
Consider the transmutation 
2 (lr aut 
Tu = Ym —— UD Ne ee rr Ka a Pes 
A mls ve 
« 
For domains («) with the origin as centre we have evidently 
a == ea, hence, 
Ot or e=4 
Yet the radius of convergence of v= Tu is not half of that of 
u but exactly equal to it, which becomes clear it is known that 
the transmutation (12) represents the operation D-!, the point 
x= 0 being taken as the lower limit of integration. The operation 
D=! is therefore a transmutation of such a kind that it produces a 
transmuted in the whole domain (a) for any function belonging to 
(a), but it is in that domain determined by a series of the form 
(1) only in so far as regards functions belenging to the circle 
(8) = (2a). The fact is that, with regard to the last mentioned 
functions, the circle ($8) for the series (12), which is not a power 
series in the letter v, forms the greatest circular domain of conver- 
gence, but not for the power series into which (12) may be trans- 
formed; this, however, is not at all unusual. From the transmutation 
oe antl 
ENE See Pe Mee nae KO) 
which is obtained from (12) by cancelling the alternation of signs, it is 
at once to be seen that it cannot produce this phenomenon so 
generally; it is clear that, in this case, for those functions u, 
70 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
