On the Radial Derivative of the Delta Distribution

Abstract

Possibilities for defining the radial derivative of the delta distribution δ(x) in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar to but different from the standard distributions. The radial derivative of δ(x) then belongs to that new class of so-called signumdistributions. It is shown that these signumdistributions obey easy-to-handle calculus rules which are in accordance with those for the standard distributions in R m .

Distributions: a spherical co-ordinates approach

“…Clearly s U (x) is a bounded linear functional on Ω(R m ; R m ), for which, in [3], we coined the term signumdistribution. Now start with a standard distribution T (x) ∈ D ′ (R m ) and let T(r, ω) ∈ D ′ (R×S m−1 ) be one of its spherical representations.…”

“…Apparently there seems to be no possibility to uniquely define the actions of the ∂ rad and ∂ ang operators on a standard distribution by singling out specific distributions in the equivalent classes ( 9) and (10), except for the following two special cases. This first special case is illustrated by the delta-distribution (see also [3]):…”

“…. , m. This problem was treated in [3] for the special and interesting case of the delta-distribution δ(x), the following spherical co-ordinates expression of which is often encountered in physics texts:…”

“…So the question arises how to explain the fact that, proceeding stepwise by derivation with respect to r, the even order derivatives of δ(x) apparently make sense, while its odd order derivatives are zero distributions, in this way violating the basic requirement of any derivation procedure that ∂ r ∂ r should equal ∂ 2 r . Let us to that end have a quick look at the functional analytic background of this phenomenon; for a more systematic treatment we refer to [3].…”

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When expressing a distribution in Euclidean space in spherical co-ordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the delta-distribution δ(x) (the angular derivatives of δ(x) being zero since the delta-distribution is itself radial) led, see [3], to the introduction of a new kind of distributions, the so-called signumdistributions, as continuous linear functionals on a space of test functions showing a singularity at the origin. In this paper we search for a definition of the radial and angular derivatives of a general standard distribution and again, as expected, we are inevitably led to consider signumdistributions.

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