ABSTRACT
This chapter extends concepts, definitions, and procedures to functions defined in three-dimensional space. As in the case of one dimension, the defining property of the fractional Laplacian lies in the relation between its Fourier transform and that of a function under consideration with reference to the magnitude of the wave number. Although most concepts in three dimensions, or any dimensions, arise as straightforward extensions of those in one dimension, subtleties requiring careful attention arise. The chapter outlines the notion of the principal-value integral and regularized integral representations. The chapter demonstrates that the fractional Laplacian defined in terms of a principal-value integral satisfies the fundamental property, and presents the Green’s function of the fractional Laplace equation.
