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Outline of a Theory of Functions of an Abstract Variable 
sphere of radius p and center xo, then it is expressible in terms 
of a power series about xo, with a sphere of convergence equal 
top. In other words the sphere of convergence extends up toa | 
point where the function ceases to be regular—an important ob- 
servation in connection with the process of “continuation.” 
Since we may differentiate a power series any number of 
times at points within the sphere of convergence, we have for a 
function f(x), regular in the neighborhood of a point x= x», 
Taylor’s formula: 
1 
A(x) = flea) + f'(u0)(x — 0) + > f'(o)(@ — wa)? + >> 
Continuation. Let>>,a Ir+-nj(X — Xo)” be a given power series 
with a radius of uniform convergence equal to pi>0. Then if | 
we have 
HO pic ee 
the function thus defined is regular throughout the whole range 
R, of its definition. Now let x; be some value of x within this 
range and consider the function fo(«) where 
fx(x) = >. { Oe (C b ) OB frpnph} (Xr — na) (x 42)"; 
The latter series has a radius of uniform convergence p2Zpr 
—||x1—xol|, so that f2(x) is well defined for a range R; that lies 
partly, at least, within R). 
The function f2(x) is called a continuation of the element filx) 
if Rz lies partly outside of R:, otherwise we obtain nothing new. 
By forming all continuations (this of course is not actually pos- 
sible, but the conception is assumed allowable in the construc- 
tion of a class) of the original element fi(x), then all contin- 
uations of these continuations, and so on, we construct an 
aggregate of functions that we denote by f(x). This function 
is said to be generated by the element fi(x). Its range is the sum 
of the ranges of the functions that comprise the aggregate, and 
its values are those of the same functions; hence, in general, 
F(x) is not single-valued. We shall call f(x) the extended regular 
function. 
T.36. The extended regular function generated by a given 
element f,(x) is identical with that generated by any among the 
