Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 92
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92
integer while q—p is a positive integer or o. q!j{q—p)!
is in this case infinitely large, and we niay put the result
aside as not serving our purposes. I have above (p. 9) pointed
out that this result is a consequence of our erroneous sup-
position that the right-hand member of (1:19) could in every
instance be given as a power of x. !n the case just now
quoted we are dealing with ordinary integration {p a negative
integer) where the lower limit of the integration is omitted
or has disappeared, x being o if q—p>o while x being co
at this limit if q—p>o. In problems where both cases occur
simultaneously, it is necessary to consider the result care-
fully. At the boundary q—p=o we may expect x of the
lower limit to lie between o and co or to be a finite posi-
tive quantity, say a. Then we put p = q——m where m is
a positive whole number and, therefore
Ð-m{x m)= Lrf! (X0_fl0)
(-l)'n~; (1+x—1)°—1
(m—1)/ o
(-i)”~;y- (-D/
{m— 1)/ ~7 n!{o — n)!
(x—1)"=
(—l)"-7
{m-\)!
n 1
(-1)
11
n—1
(x 1)
n
{-\)m-J
(m-\)!
lnx
(49)
Now, if r and m are positive whole numbers, r>m, we may
interpret
Ð~r{x-m)
(~m)!
(r—ri) !
xr~
m
of our general formula (1 :19) as
\m—l
lnx
(—l)m-/xr-m /, 1
(m—\)!(r—m)! ( n,X n
' n = 1
(49a)
If necessary we may use this interpretation of the end result.
We shall now proceed to treat the problem of obtaining
z from the serial relation