On fractional calculus and fractional multipoles in electromagnetism

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 an extension to the 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+α)P_α(-cosθ) where 0<α<1 and P_α(·) is the Legendre function of noninteger degree α, thus denoting this charge distribution as a fractional 2^α-pole. At the two limiting cases of α=0 and α=1, this fractional 2^α -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^α-poles are also given