1228 
This series of fractions represents p‚(z), if |2|} <1, s > 0, on the 
other hand it is equal to p‚(z) as soon as | zl > 1 and at the same 
time s >1. We now can apply a theorem due to Goursat *) and 
conclude that the points a, without exception are singular points 
of the functions p‚(z) and p‚(z). Hence, as these points form a set 
dense on ©, the continuation of either of the functions across the 
circle is excluded. *) 
By application of EtLer’s summation-formula we can calculate 
the values taken by the functions p‚(2) and g,(z), when z along 
the radius approaches one of the singular points. In this way find 
in the first place, when z has a positive value a< 1, the following 
asymptotic expression for p‚(2) 
gin ae ~$(s +1) + (toe ‚) I’ (1—s) § (1 —s) — § §(s) + 
log— 
a“ 
; B, 3 6 
ee 2 log - = fs S(s—1l) — — (tog, ) §(s—3) + — 5 (los - - ¢(s—5) —.... 
holding for all en values of s. 
The result is less simple, when z tends along the radius to the 
ong 
point e 9 ef. Putting z— ge'®, | get for 0 < 1 and supposing 
again s to be a non-integer *) 
') Bulletin des Sciences Math., t. XI, p. 109. Sur les fonctions à espaces lacunaires. 
n == 00 zn 
2) This results also from one of the propositions concerning the series X /), Tb 
n= EE 
enunciated in a previous communication (Verslagen en Mededeelingen. XXVIII. 
p. 269) according to which the continuation of the function across the circle is 
impossible, as soon as 6, >0 and Lim b, —0. 
n= 
3) For integer values of s the result is obtained by making s tend to the 
integer limit. So for instance, if s tends to zero, we ‘will find 
1 
C — log log — 
wv 
; B. 1 ig avs 
y{e): = î rg Sng Nas Pd En log — ahs lair 
log — 
wv 
and 
Oe log | 1 
nape FOB 7 108 °8 | Rn 8 
Lim {p, (ge?) — = met = heot—. 
p> 2q h=1 2 
1 
q log— 
e 
The former of these formulae was obtained by ScHL6MILcH, the latter | deduced 
in a previous paper: On LAMBERT's series. 
