1102 
which are identical with their natural majorants*) it holds that 
r= 4,7 and r are the corresponding radii of convergence of v = Tu 
and u. If we further take into account that for the transmutation 
(13) we have 
Tue tua = Due), 
it is to be seen that the same may be said of all functions u. 
The operation 
7 Ee 2m (m) a 
C= m EN Pay te wee aye 
(m +1)! : Gg 
in which a(a)= a’, and therefore 
=a =e, Or a= iV PLT = i, 
gives for majorant functions transmuted for which the corresponding 
radii of convergence 7” and r depend on each other as a and 8, 
but for the function 
=S 
? 
14e 
for which » =1, a transmuted 
4 
1 a? v 
fig Te 
se! ls 
o— 
or 
| 
ltete? j 
which also has 7’—1 as its radius of convergence, while, however, 
the series (14) only converges within a circle with radius 
e=1(V4X14+1—)N=}3(y5—1=—06... 
Here again the example becomes clearer by the meaning of the 
transmutation, which in this case appears to be a substitution, viz. 
FS ard) ee ene 
If in general 
LH Sula) ; 
we find in the memoirs of BourLeT and PINCHERLE 
1) J. TANNERY, Introduction à la théorie des fonctions II, N°. 343, under- 
stands by the natural majorant of another function, the one that is deduced from 
the first by replacing the coefficients in the development of its power series, in the 
neighbourhood of the origin, by their modulus. We shall, if there is question of 
the neighbourhood of an arbitrary point zo, understand by the natural majorant 
of u the function obtained from wu by replacing the coefficients for the point xy 
by their modulus. 
