Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 4

4
side of (1) the respective Maclaurin series. However, to
facilitate future applications it is worth while to consider
how (1) can be obtained from the general serial relation4):
dy . P2 2 d2y
p0y-Pix~ + ~x —2
dx 21 dx
/j3 3 d3y
X
do'Jo + CílX
d'Jo
dx 2!
3! dx
q2 2 rf2//0 ,q3 3 yQ
X , 2 "T X
dx 3!
3 H— • • •
i3 í/„
+ ...
dx3
(3)
where the following relations exist between the coeffici-
ents p0, pv p2, p3, etc. and í70, qv q2, etc.:
or
- «!(-ir
m=óm\(n — m)\
~ n! (- l)m
7öm\(n — m)\‘
P n
dn
(4a)
(4b)
We have at our disposal all the coefficients q0, qv q2i
etc. while the p-coefficients are then uniquely determined
by (4a). However, we could as well choose the values
of p0, pv p2, etc. at your will. The q-coefficients are then
wholly determined by (4b). Obviously we could also select
at our convenience the values of some of the q-coeffici-
ents, and some of the />coefffcients, provided that the total
number of the coefficients thus disposed of up to any given
index n does not exceed n + 1.
Usually we put q0=í—p0 and of the remaining 2n
coefficients we have t-hen n free at our disposal. This con-
dition must be fulfilled not only for a given value of n
but for any lower value.
The coefficients (p and q) may contain the independent
variable, x, if they satisfy the relations (4a) or (4b).
Now we proceed as follows: We attach such values to
the coefficients at our disposal that the sum of the first