Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 17
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j(x). On the other hand the series Ai to AG are valid although
cp(i'i) contains x, whereas it is assumed that the symbols
Ðx and Ax (or Ðinx) do not operate on x if contained in
<p(r|).
The transformation froin A to B is important, but it must
be kept in mind that its validity depends on the supposi-
tion that qp(r|) does not include x.
A3 is only a special case of A2, a==—1. On account of
the fact that it is frequentiy applied, I have considered it
worth while to put it down separately.
The series Ai, A2 and A3 are based on the supposition
that y is developable according to Maclaurin theorem. If not,
then we can transfer the origin to another point or, what
is often still better, we can put
TWl + Ari/m))~> <p(») =X"(\+ cp(n)
KxO + a^AO) / , -
X x*(l +Av/a)* W
and deal with ><(x)x"-9 instead of y(x) in one or more of
the series Ai,A2 and A3.
Then we get
_p(x(l+Aí7/co))-xp(n)
=~ cp(n+co(<7+/?))
n—0
n=U
cp(n+co(^—n))
n=0
dn(y(x)x q)0 X<?+n
dxn n!
vv dn(y(x)x-% ® n
’’ d(xa)" n!
dn (y(x)-x-q)o xq-n
d(l/x)n n!
Ai1
A2‘
Aa'
However, the series Ai, A2 and A3 are still valid accord-
ing to my view, whereas they are in such cases not suit-
Symbolic Calculus — 2