Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 100

[5]
Here I have put
100
w(x) = \ (n-\-h—k)(n-\-h—ki)Pr,
n-\-h
dn ( d"y-\
l dxh l0 xn
dxn
n!
dhy
— (Ðlnx~\-H—k){Ð\nx~\~h—k\) Pr,Ðlnx-\-h —> dxk (öl a)
From (61) and (61 a) we are able to evaluate the error com-
mitted by dropping the remainder Rh,r.
r—2
For h=k, (61 a) becomes
w(x)=XÐx{Ð\nx~rk—ki)Pr,Ð\nx+k ~
dky
dxk
(61 b)
dky
(61 c)
and ii we have also k\.,=kj\
w(x)=X~Ð2x Pr,Ð\nx+k —r ^xk
We observe that the formulae (61) and (61 a) are, with
obvious alterations, generally applicable to evaluate the
remainder Rh of the series
_ ^ZL dny0 xn , dny0 xn , _
2 dxn n! 21 /(«) dxn n! + ^h'
n=o n=o
The differential equation (55) may also be written as
follows: —
m m
^(jc-FÐo) -> g(r) JCr-or=j/(jc(l+Ð0))
g(r).or
=e oy(xÐ0)=>e°2__g<r>or=y(xÐ0)-*e°^_ g(r)or (62)
This transformation is based on (3:11 a) where q=—1.
The symbol Oi means zero but the index shows that Ð0
does not operate on it, the operation of Ð0 being restricted
to o, the other vanishing quantity. We observe that
Ðn(ore°) =
n!
(n—r)!
and (62) leads, therefore, immediately to (56).