1104 
Tu = 8,11 (u). 
Here a; = w(«)— «=1, consequently a=1. For the function 
1 1 
u— — ‚we have 7u— 
l+e 2+ 2 
the resulting function exists in every domain round «=O with a 
radius smaller than 2, while it is not determined in any small 
neighbourhood whatever by a series of the form (1), which is here 
oe —]\m+1 
7 lte 
That it is quite common if the transmuted function does ezist 
in some domains, but is not represented in it by a series of the 
form (1), appears moreover quite ciearly as follows: For a definite, 
given function the domain may be constructed once for all, where 
the series (1) produces anywhere a transmuted for that function as well as 
the domain where it produces nowhere a transmuted. Consider for 
this purpose all singular points s of the function and construct to 
each the set of points with the property that the value which the 
magnitude a, has there is smaller than their distance to s, and 
also the set of points where that value is greater than the 
distance in question. The latter set of points shall as a rule consist 
of continua round the singular points; in none of those continua 
the series will converge, but it is clear that in general the trans- 
muted of the function considered may be continued within those 
continua, except in a few points which may but need not 
coincide with the singular points. Thus in the transmatution (12), 
which answers to D-—!, and where a; == |z|, the domain of points 
where a, is greater than the distance to the point z=—1, is the 
continuum on the right of the straight line «= 4, and the series 
(12) does not converge in any point of that half plane for 
functions which only have the singular point z=1. The operation 
D-', however, which is represented by the series in the half-plane 
on the left of that line, exists moreover anywhere in the first mentioned 
continuum, except in a line starting from the singular point. In 
the example of the substitution S,4, which we gave last, we have 
A; —=1; hence a function with the singular point «= —1 has not 
in any point of a circular domain with that point as centre and 1 
as radius, a transmuted determined by (1). Nevertheless it has a 
transmuted function, except in the point #—W—2 of the circle 
mentioned (here the singular point has been displaced by the trans- 
mutation). For the substitution S,.) in general we have for the point 
«= 0,dr= |w(0)|. If therefore the radius of convergence of u is 
. 
’ 
