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Fractional Analytic Extensions of Multipole Expansion via Directional Derivatives
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DOI：10.1002/mma.70713.png)
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Abstract
En 中文
We develop a rigorous framework for Gaer fractional directional calculus, providing a complex-analytic extension of classical differentiation and integration along arbitrary directions in . Starting from Gaer's contour representation of integer-order directional derivatives, we construct a fractional operator that extends differentiation to complex order while preserving analyticity, decay properties, and geometric invariance. The resulting operator forms an analytic family with respect to the order parameter and unifies several classical fractional constructions, including the Gaer, Weyl, and Riesz formulations, within a common analytic framework. Building on this structure, we derive a fractional extension of Maxwell's multipole expansion. By analytically continuing the differentiation order from integer to complex , we obtain a generalized Maxwell-Legendre formula in which the Legendre polynomials are replaced by Legendre functions of complex degree. Applied to the Newtonian potential, this construction produces a continuous family of fractional multipole potentials that interpolates smoothly between the classical monopole, dipole, quadrupole, and higher multipole fields. The resulting theory establishes a natural analytic bridge between fractional calculus, harmonic analysis, and potential theory, and provides new tools for the study of nonlocal field models and fractional generalizations of classical electrodynamics.
Keywords:
analytic continuation
fractional calculus
fractional directional derivative
Gaer fractional derivative
harmonic functions
Maxwell multipole expansion
nonlocal field theory
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M
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1.8
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605
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