1233 
Determining then the integer & by the condition (ns — 4’ < 4 and 
putting z= oe2*%, we get by the same reasoning as before 
1 
n> ’ if cos 2 ans < 0, 
1 | | ke) 2n ' 
Ent ‘jens ; £0, 
= 1 >a § n\> Gp let ‚if cos 22n§ > 
and consequently 
1 en | 1 ‘ 
tamer ag Gos 
nl len, nl 
1 ze | (/+ 1) 1) 2n+1 : 
ae: ial ie f ed 
nl 1—2"| On. nt ‚if cos 2 us > 
Hence the series G(z) will converge absolutely on the radius of 
the point e?** and the convergence will be uniform on any segment 
of that radius. 
Thus then, we have shown that in ‘this case the functions y, (2) 
and w,(z) are connected at all points of an aggregate of power c 
and that along the radii of these points the series of Lampert G (2) 
procures a faultless continuation, whereas the analytic continuation 
necessarily fails *). 
The elementary examples I discussed show as well as the examples 
of Borer that sometimes we are led to regard as a single function 
a group of distinct analytic functions existing in separate regions. 
And from the fact that in these cases a non-analytic continuation 
can be effectuated, the question arises whether a certain extension 
should not be given to the concept of functionality. Boren, made a 
step in this direction by developing the theory of a class of non- 
analytic, monogenic functions existing in a so-called domain of 
Cauchy °). 
1 1 ; 
1) As we have — < — for all yalues of s, if only x is sufficiently large, we 
nl ns 
are certain that the series G(z) also furnishes the continuation along the radii of 
algebraic points of order whatever. 
*) Lecons sur les fonctions monogènes uniformes d'une variable complexe. 
Chap. V. 
