Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 10

10
For me it is important to emphasize that in the calculus
discussed here, there is no intrinsic diffenence between the
factorial «/and the gamma-function q! in that ///—(1 -|- a0)"(0!)
— the factorial, and q!—(\4- Ao)7(0!) the gamma-function,
where n is a positive whole number as usually, whereas q
may be a fractional, irrational or a complex quantity. As to
imaginary and complex quantities we shall, however, discuss
tliem very little in this paper.
My symbols a.v and Ðx frequently operate on x when it
is in the state of vanishing or .v=0 for the sake of brevity
it is tlien often practical to write a0 and Ð0, for instance
A,
'1+.V
and Ðn
x=0
1 -f 0
or
"1+0
do
for
“ 1 +.v
dx x=o
When used in this manner o becomes a specific zero
that ought not to be interchanged with o derived in a dif-
ferent way. To avoid mistakes we sometimes add an index
to the specific zero to remind ot its specific function.
The use of the Symbolic Calculus, especially in connexion
with the Serial Relations, facilitates very much the mathe-
matical development. We must, however, rernember tliat the
Symbolic Calculus rests on certain assumptions, its irreproach-
able use implies, therefore, a strict observance of the rule
that we shall in every case remain within the boundary thus
established.
The mathematical manipulation of the symbols A.v and
Ðx as if they were ordinary quantities, is allowable only if
there is no x mixed up with them. For instance, it is not
permissible to equate x" Ð'ý and (xÐx)" and, therefore, the
infinite series
,n n
-d y x
ndxn
n 0 /i-O
/? r\n/ \
x Ðx(y)
n!
need not in our Symbolic Calculus be equalent to
y-(xÐx)n(y)_
/i—0
nXÐ-
n/
(y)