Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 3

1. THE GENERAL FORMULAS
From the point of view of a physicist it is important to
have recourse to a general method for solving differential
equations without regard to their special forms, although
the results obtained may be only approximate. For this
purpose I have proposed a method which I have called
Approximative Integration1), but it seems to me now that
asymptotic integration is a more appropriate appellation.
The general formula may be written2):
r\ (r+s — «)! (— x)n d"ij
n — o (r — n)! (r -f- s)! /r! dx"
S
s!(/- + s — rí)\xn
(s — n)! (r -f- s)! n!
dny0
dxn
Rr,Áy)
(1)
where
d\
dxn
denotes
x= o
while the remainder term Rr s (y) is a sinall quantity, espe-
cially when r and s are large numbers and x not too great.
When r = s we get the principal formula:
s!(2s—n)\x"
(s — /i)! (2s)! n\
(2)
Obviously (1) is reduced to Maclaurin series if r — 0.
The deduction of (1) can be accomplished in various
d”y
ways.3) We may e. g. substitute for — on the left hand
dx