Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 7
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[1]
and
(1 +Ai/<o)"/(x-)=/(x+«a)) (1 Oa)
where n is a positive integer. On the other hand the Taylor
theorem yields
eno)Ðx =/(.v-)-(,)/?)
and therefore
(1 + Ax/m)'—en“ftv=(e“ Ðv)"
or
1 hA.r/o)=e,’,av (11 a)
and for o>=l
l+Ax=eÖA‘ (11 b)
According to our assumptions we conclude from (llb)
(1 +Axv' —epÐx
valid for all values of p and therefore generally
(l + :\v)'7(.v)=e'Æv/(.v)=/(.v+/+ (11)
(1 +Ax/»)" f{x)—el> '"°v /(x)—f(x • /<■»/;) = (1+A >'’/(*),
1 +Ax/<o=(l +A_V)‘,> (12)
By means of (2) and (l) we readily obtain:—
2Z«/7v+/2j;A;r//,"==(l+ A+)"
and thcrefore "=o
;; '{p fjJ I -+ “/W (1 "I- -+ ,',’),J f(X) f(X “I- ("/?)
(13)
which is Newton’s interpolation formula. According to our
conception of the infinite series and with due reservation re-
garding certain periodic functions as mentioned above, tliis
formula is always valid.
Hitherto we have considered /; of as a whole num-
ber, we are now, however, able to drop this reservation. If
p is positive, the following transformations evidently liold:
a'!=(1+Ax-1)" =
P-
n 0
n!(p—n)!
,(l+Ax)"-"(-l)"
n 0