On fractional calculus and fractional multipoles in electromagnetism

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 distributions are concerned, such fractional-order multipoles effectively behave as ''intermediate'' sources bridging the gap between the cases of integer-order point multipoles such as point monopoles, point dipoles, point quadrupoles, etc, This technique, which involves fractional differentiation or integration of the Dirac delta function, provides a tool for formulating an electric source distribution whose potential functions can be obtained by using fractional differentiation or integration of potentials of integer-order point-multipoles of lower or higher orders, As illustrative examples, the cases of three-dimensional (point source) and two-dimensional (line source) problems in electrostatics are treated in detail, and extension to time-harmonic case is also addressed, In the three-dimensional electrostatic example, we suggest an electric-charge distribution which can be regarded as an ''intermediate'' case between cases of the electric-point monopole (point charge) and the electric-point dipole (point dipole), and we present its electrostatic potential which behaves as r(-(1+alpha))P-alpha(-cos theta) where 0 < alpha < 1 and P-alpha(.) is the Legendre function of noninteger degree alpha, thus denoting this charge distribution as fractional 2(alpha)-pole, At the two limiting cases of alpha = 0 and alpha = 1, this fractional 2(alpha)-pole becomes the standard point monopole and point dipole, respectively, A corresponding intermediate fractional-order multipole is also given for the two-dimensional electrostatic case, Potential applications of this treatment to the image method in electrostatic problems are briefly mentioned, Physical insights and interpretation for such fractional-order 2(alpha)-poles are also given.