N. Engheta, ON FRACTIONAL CALCULUS AND FRACTIONAL MULTIPOLES IN ELECTROMAGNETISM, IEEE transactions on antennas and propagation, 44(4), 1996, pp. 554-566
In this paper, using the concept and tools of fractional calculus, we
introduce a definition for ''fractional-order'' multipoles of electric
-charge densities, and we show that as far as their scalar potential d
istributions are concerned, such fractional-order multipoles effective
ly behave as ''intermediate'' sources bridging the gap between the cas
es of integer-order point multipoles such as point monopoles, point di
poles, point quadrupoles, etc, This technique, which involves fraction
al differentiation or integration of the Dirac delta function, provide
s a tool for formulating an electric source distribution whose potenti
al functions can be obtained by using fractional differentiation or in
tegration of potentials of integer-order point-multipoles of lower or
higher orders, As illustrative examples, the cases of three-dimensiona
l (point source) and two-dimensional (line source) problems in electro
statics are treated in detail, and extension to time-harmonic case is
also addressed, In the three-dimensional electrostatic example, we sug
gest an electric-charge distribution which can be regarded as an ''int
ermediate'' case between cases of the electric-point monopole (point c
harge) and the electric-point dipole (point dipole), and we present it
s electrostatic potential which behaves as r(-(1+alpha))P-alpha(-cos t
heta) where 0 < alpha < 1 and P-alpha(.) is the Legendre function of n
oninteger degree alpha, thus denoting this charge distribution as frac
tional 2(alpha)-pole, At the two limiting cases of alpha = 0 and alpha
= 1, this fractional 2(alpha)-pole becomes the standard point monopol
e and point dipole, respectively, A corresponding intermediate fractio
nal-order multipole is also given for the two-dimensional electrostati
c case, Potential applications of this treatment to the image method i
n electrostatic problems are briefly mentioned, Physical insights and
interpretation for such fractional-order 2(alpha)-poles are also given
.