Rit (Vísindafélag Íslendinga) - 01.06.1951, Síða 6

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The common denotation
A/(X)-/(.V-(-«)• f(x)
will here be replaced by
ax4)->/(.v)= a^/W^/^^ +/)) -/(v)=/(x+o»)~/(x) (8)
As is well known the syinbols mentioned can be treated
in mathematical formulae as if they were numerical quanti-
ties, and, as (6) and (7) show, Ð" and a" are reproducible
in ordinary mathematical language if n is a positive integer.
If n is a positive integer, we have the relation:—
Ð" • Ð~n—\
Ð~" corresponds, therefore, to a «-fold integration where x
is the upper limit of the integrations, while the lower limits
are constants to be determined in accordance with the prob-
lem involved.
Similarly
A«. A-"=l
and accordingly
A x"f(x)=f[{x) (9a)
if
A"<p(x)=/(x) (9b)
The function cp(x) as determined from f(x) of (9a) includes
n (arbitrary) constants. Further any constant included in tp(.v)
may be replaced by a periodic function in x the period of
the function being 1. We exclude, however, all such perio-
dic functions from our discussion in that we postulate that
the general solution shall not be affected by changing a*
into Ax/w where o> is an arbitrary constant. At first glance
this restriction may seem to be very serious. Actually it is
of no importance except in very special problems. On the
otlier hand it is highly helpful in general problems as it
puts Ai on even terms with Ðx.
From (7) and (8) we readily derive
(l + Ar)''/(.v)=/(.v | /?)