Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 15

15
'A+i—c) 1 - («0+«i(Ai—c))(x «g(^i - c))(x—AQ8+.. ■
M'-l- c)+{fio+Wl - c))(x - h) + (Pl+Mh - C))(x- Al)s+...
then we are able to arrive at a solution of the problem by
a similar procedure to that above described; we have only
to substitute «„ (^i—c) + «„-i for «n and Pn(h—c)-\-pn-i for
/?n. Obviously this procedure can be extended to the case
where y does not at all vanish at the limits of integration.
When the limits are x = ci and x = c2 it may easily be
shown that we can substitute
(+—Ci) (+—c2)an + (2Ai—ci—c2)an-i + a„-2
for «n and, similarly
(+ — Cl) (Ai — C2) + (2/l — Cl — C2) /ín_ 1 + fin-2 for fin,
We shall now proceed to illustrate this method by a few
examples:
F. Let p dx dx and y vanish at one of the limits
y * fio
while the other limit is x = c.
Then we have:
dy «o(Ai —c) + ao(x—Ai)
dx.
y /^o(^i—c) + +(x— Ái)
From the relations (16a—e) we gather:
«o(Ai—c)+/?o 0, /j«0+/5o(Ai—c) = 0, A3a0+2/2//o = 0 etc.
whence h—c = í 1+2 or c = Ái + //2
«0 4-1 - 2
and
/3 —
and so on.
t*o Vh ’ P'
We could have obtained the same result by integration:
/’CVJ re( p+»/)c
ev C / e(p+'',I< dx = — ---------r—
J c P+’?
C (c-l),/++ _ + +■
— e p / 2p 3pj
Hence
/1 = c—
x =1
2 p8’
2
P8
/3 = — r§ etc<
which is in complete agreement with our former result.