1232 
Again some insight in the behaviour of these functions in the 
neighbourhood of the singularities is obtained by the application of 
Evrer’s summation-formula. Giving in the first place z the positive 
value «<1, I find 
1 B peep 1 
wp, («) = —— {Li@) —Cj—4e—) +9) log — 7% -5e( log ) + 
1 24 i U 4! U 
log — 
v 
B, EAR 
+- 6! . 52 é (toe) a” Oy le „js 6 
and the absolute value of the error committed by stopping at any 
particular stage in the series always will be less than that of the 
last written term. 
Ey} 
osb 
Putting then z= ¢e 7 — oe? and making @ tend to unity, we 
will find 
; il fe 1 h=q-1/h, te 
Lim jw, (oef) — DT = ENT =— 4 (el) Ss (<- 5 oats 
el Des (kg) ! kg h=1 \Q 
The function w,(z) behaves in the neighbourhood of a singular 
point in a similar manner because of the relation 
(2+ w. (=) =d (el 
Now, let § be a transcendental number of the interval (0,1) the 
expansion of which in a continued fraction gives 
FD atid WR 1 
ata, da, ae ak 
where all integers ay, are less than a given finite number J. 
Evidently these numbers &, and therefore also the points e?™ 
form a set of power c, the set of points e?* however being not 
dense on the circle. By the known properties of continued fractions 
. . . Al 
we have, & being an arbitrary integer, N the n-th convergent 
AN n 
| k | | Th Tne Ta 
An+-2 1 i 1 
eas (aneNi + Nn) > ZN No es an + IJN’ ee Bi aN 
and as N, is manifestly always less than (/-+1)", we may write 
k iL 
Fae 
