Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 20

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convergent series, does not hold because we can readily
transform the divergent series to asymptotic series by a
procedure which I have described some years ago5).
Let
U = «0 + «!* + a2x2 + .... = y~ a„ x" (30)
n =0
1 d"y0
where a =---------- is a known function of n.
n\ dx
We have
(/j0+p{x+p2x2 +... + prx) y
= b0 + bxx + b2x +... = ^Jbnx" (31)
/1 — 0
where
hn = anP0 + an-\Px + an-2P2 + + an-rPr (31a)
The constants p0, />,, p2....pr have to fulfill the condi-
tions ös + 1 = 0 = 6s+2 = ös+3 = ... = öj(+r. p0 is chosen
conveniently and the remaining px, p2,...pr are then readily
found from r linear equations (31a).
Then we have
b0 + bx x + V2 +.. . + bsxa + RrJy)
P0 + PxX + P^~+ ..' + prxr
(32)
where the remainder:
Kjy)
_ X' + 1 /'/J _ rfS + 1 ([Po + Pl®x + P2 (l<>X)2+ •+Pr( í)x)'] y («)//)) J
s'. j 0 ' d(í)x) s+l
21
*r+*+1 ^ ,+r ++*+'í(Po + P+x + ... + pr (hx)r) y («x)]
, '’+s+l
(r + s)L
we can also put
where z satisfies the differential equation:
(P„ + P, + p2*2 + • • • + P+0 //
— r! (r + s — /;)! (—x)" d"z
d{)
(33)
(34)
(r— n)!(r + s)! //! dx"
(35)
However, in practical calculations it seems to me more
straightforward to consider the remainder as an infite series
to which I apply the same procedure as to the original
series in order to acquire sufficient knowledge about its
magnitude.
Making use of determinants we are able to give the
solution of (32) in a very lucid form:
*+i
* — í
.... x
... .0
•+2“. + l
. a
s-f-1 — r
s-\-2-r
» + r öS4-r_i «s+r—2 •
n — 0
an an-1 °n-2••• ■an-r
«., + 1 °s «,-.••• • •«, + !-
as+2as+X as • °s + 2-

a s + r a s + r— as+r- 2 • ••«,
/i-0
a„ a„ °n-r
a , . a a ,
us + l
a , o
us+2us+l

«s + ras + r-l • • • • «,
(32a)
x” + Rr<x (y)