=]2=
is a convergent power series in R,, it is also absolutely conver-

" converges absolutely in the same region, ¥

gent, and therefore v¢
Therefore the terms of (5) can be arranged accerding to powers of
%, for each value of #, and the resulting power series is absolute-
ly convergent. After this rearrangement is made, (4) becomes a
uniforrly co-nvergent series in R,, évery term of which is a power
series convergent in Ry, A theorem by Weierstrass states that such
a series can be arranged according to powers of x, the coefficients
of the various terms of a given power being added to form a single
coefficient of that power. The }resulting series converges absolute-
ly for |x| < Ry. In making this last rearrangement, we find that ip
the terms of (4) the sare power cf x ray appear in several powers
of Ȣ. To collect all the terms containing %", we must pick from
the several powers of vi; that is, we allow » to take en several
values while h is kept fixed, In picking out all the terms having
<", we get [ %—’:,"—' a;? b® *e+ 1,1, the summation being taken
for the values cf j,"k, ***, r, which satisfy the equation
ja+ B4 +rA=h, (here having a changing = which by definition
is »=j+k+ *"*4r-1). let u,, denote the expression inside the
brackets,

Now we can write log [1 4 v4] = ujex 4 U X° 4 Ugex® 4 ceeves
Since this is a convergent power series, each uyp represents a
finite quantity; There are as many terms in u,p as there are solu—
tions of the equaticn %+ 84 4rh=hn, This shows that Uy
has a finite ‘nu‘mber of terms for each finite 4. Now write
: u=3 dog(1 + vy
Br ligaX + Uggx? + ugqx® 4 to00es 4 ugpxh 4 eeees
(8 g R R S o S R R

_““’, + ug,xa + "lsxs 4 ccecee 4 "{h"h 4 tecee

80 Bromwich, sec, 27. Smail, sec, 146, 165.