587 



Again it may liappeii that (fit) can lie divided into tlie sum of' 

 two functions r/,(/) and '/,(0, llie tii'st of wljicii is regular at / = 1 

 and the second of whicii has the latter point as its only singularity 

 on the circumference of the circle (0,1). if, then, the number //for 

 (Pi{t) is equal to P./ and that for ^^^(i^) to P./ and if A/ ^ ;./, so 

 that for the whole function r/(/) the nund)er I' is e(pial to ?./, the 

 integral (1) can be expanded in a conditionally converging series of 

 factorials for 



r, + 1 < Ri^v) < ;.' -j 1 



if A' = ;./ < A/ -}- i. and for 



;/ < «(.r) < ;.' 4 1 



if A/^A/-j-l. If, in this case, (/,{t) has the properties of Hada- 

 MARD, then P./ -\- 1 = X^ = )., and the pioposition of Niklsen is 

 valid, which, now, really has a particular meaning. 



7. The following proposition is, as a corollary, included in the 

 theoi-em of the preceding paragraph : 



// the coefjicients a„ of a function <f{t), defined by a /)owey-serie.s 



00 



ff{t) — Vl ant" 

 



which has the circle (0.1) as its douïain of convergence, are, for 

 71 = 00 , equivalent to a power n^' of n, and if the series 



eo 



B-^ '25) 







is divergent for 0<^<^1, the point t:=\ is a singularity oUf{t). 

 For if ^ = 1 is an ordinary point of 7(0, the series (6), which, 

 except for the factor P{x), is equal to 



V» n! a„ 

 n (26) 



^ n.rfn+1) 



is convergent foi* R{.i') ^ ;/ and the convergence of (25j can be 

 derived from it. For we may write 



7i! a„ n! a„ r{x ^ ti-\-\) 



7,^'i ~~ rfr+n f 1) Tin i l)n>'-|-« 

 If we choose .r such that A' < /t! (./■)<-?■' + é*. the series fornied 

 by the tirst factor, if // takes all values from zero to intinite, con- 

 verges, as we have already seen, whereas the series, composed of 

 the terms obtained by taking the first Jinite differences, with regard 

 to n, of the second factor, converges absolutely; and it is a well- 



