Abstract
Our main aim is to present the value of the distributional derivative ∂͞
N
/∂
x
1
k
1
∂
x
2
k
2
. . . ∂
x
p
k
p
(1/
r
n
), where
r
= (
x
1
2
+
x
2
2
+ . . . +
x
p
2
)
½
in R
p
,
N
=
k
1
+
k
2
+ . . . +
K
p
, and
p
,
n
,
k
1
,
k
2
, . . .,
k
p
are positive integers. For this purpose, we first define a regularization of 1/
x
n
in R
1
, which in turn helps us to define the regularization of 1/
r
n
in R
p
. These regularizations are achieved as asymptotic limits of the truncated functions
H
(
x
– ∊)/
x
n
and
H
(
r
–∊)/
r
n
as ϵ → 0, plus certain terms concentrated at the origin, where
H
is the Heaviside function. In the process of the derivation of the distributional derivative formula mentioned, we also derive many other interesting results and introduce some simplifying notation.
Reference7 articles.
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3. Gel'fand I. M. & Shilov G. E. 1964 Generalizedfunctions vol. I. New York: Academic Press.
4. Jones D. S. 1982 Generalised functions. Cambridge University Press.
5. Kanwal R. P. 1983 Generalized functions theory and technique. New York: Academic Press.
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