228 
RADELFINGER. 
valid throughout the domain of existence of the function— 
that is, throughout C. 
Several notable advances have been recently made in the 
solution of this general problem, and of these I shall give a 
orief review. 
Borel* in 1896 deduced the following expression: 
FK(x) = Lim «-* 1 *1 [> («) + (x-a) 
k ~ co A= o l_ LL 
F (2) (a) 1 1 
+.j"2 ~ 0 ~ a ) 2 . ^ ( x ~ J 
for FK (a), which converges within a region K, which is a 
polygon circumscribing the circle of convergence of the 
corresponding Taylor’s development, and therefore includes 
a larger region, and is, theoretically, a step in advance. 
Mittag-Leffler .—By far the most important results are due 
to G. Mittag-Leffler, and these have appeared in four separate 
memoirs or “ notes ” f in his journal, the Acta Mathematica, 
and which will be referred to as the first, second, third, and 
fourth notes respectively. 
First Note —In this first note Mittag-Leffler defines a re¬ 
gion, A, termed a star, which can be constructed by taking 
an analytic point a in the x plane and drawing vectors from 
a through all nonanalytic points; if the portions of the 
vectors lying beyond and including these points be excluded 
the star A will remain. 
* Fondements de la tlieorie des series divergentes sommables. Jour¬ 
nal de Mathematiques, series 5, tome 2, 1896, pp. 103-122. 
t Premiere Note.—Sur la Representation Analytique d’une Branche 
Uniforme d’une Function Monogene (Acta Mathematica, 1899, tome 23, 
pp. 43-62). 
Seconde Note.—Same title as above (Acta Mathematica, 1900, tome 24, 
pp. 183-204). 
Troisieme Note.—Same title as above (Acta Mathematica, 1900, tome 
24, pp. 205-244). 
.Quatrieme Note.—Same title as above (Acta Mathematica, 1902, tome 
26, pp. 353-391). 
