Rit (Vísindafélag Íslendinga) - 01.06.1946, Síða 16

16
Substitution from (20) gives
n .. m
m
x g
—n i n
k\
n\
n = o «! dx 9 = 1 \(k — q)\ (n — q)\
xm-qAfl
X
k\
-xm~qA
9=ö (k q)\
(23)
It is easy to see that the denominator of (23) is very
significant for the differential equation (22), and its relation
to the indicial equation is quite obvious. However, I shall
not here dwell on these questions as they are known from
other sources.
The series in (23) is in the sense asymptotic that the
,k
d y0
term containing —^ has disappeared. We confine our sum-
dx
mation to the terms where n<k. Sometimes, however, we
do not attain sufficient accuracy in that way. We have
dky0
seen that more accuracy is obtained if besides —k also
dx
dk+1y0
the subsequent derivatives — k,.
dx "l”
d í/o
dx
k + 2
etc. or the
preceding
dk~'y o
dx' ~1
etc. also disappear from the series. This
may be accomplished in the following manner. As abbrevi-
ationlwrite /= 0 for the differential equation (22). Further
m df
f = a — -f- bf where a and b are known and a^O,
dx
d2f df
öj —j --------h cj and so on. Now I
dx dx
furthermore /
(2)
apply the asymptotic differentiation (20) or (19) to the
differential equation:
'(2)j-%/(3)+ .... = 0
/ + %/(,, + %/'
(24)