S61 
function with a radius of convergence less than a. Let 
BNN AEN == ss Swe (By =, 0, 
and 
2 ):—1 (a) = ¢2k—-1, 
in which by e a constant is meant greater than 1. Here the number 
a is equal to c, hence, according to the supposition, greater than 1. 
The function 
1 
las 
for which, in the point «=O, the radius of convergence is equal 
to 1, has, though this radius is smaller than the number a, in that 
point a transmuted, which is equal to zero. 
) 
4. Bovrter’s theorem informs us about the question when the 
series (1) is complete i one definite point x,, but, as a matter of 
course, only those cases are interesting, in which a certain circle 
(o) with centre «, is to be indicated, such that the series (1) produces 
for all functions belonging to it a transmuted not only in the point 
x,, but in all points of a certain domain (a) round «,. Bourrer 
however has not drawn attention to this. If the series (1) satisfies 
the new condition, we shall call the transmutation determined by it 
complete in the domain (a). 
Let us suppose that for any point of a domain («) a number a, 
as mentioned above, can be indicated, and that the number 77-, in the 
condition 1 at the beginning of § 3, remains below a fixed number 
in the whole domain («), which number we will also indicate by 
m:. This latter supposition we shall quote in future as “the uni- 
formity supposition of N°. 4”. As the number a will in general 
depend on the place of the point «2, we prefer to represent it by 
a,. According to this supposition the expression 
i 
i a Boe |g (ey fh ee ees eed ee) 
M= 
has in all internal points and points on the circumference of («) a 
finite value. We further suppose that the quantity a, remains in 
the whole domain («) below a fixed number (, in other words, 
that a, in the domain («), is a /imited-function of the points of that 
domain (always the circumference being included). The numbers a, 
have then in the domain («) an upper boundary, which we will 
represent by a(«) or briefly by a, and we now assert that the 
upper boundary of the a,-values for points of the e/reumference of 
‘a is equal to that for the whole domain (a). Suppose that this were 
