Fractional Integral and Derivative of the 1/r Potential

We calculate the fractional integral and derivative of the potential $1/r$ for all values of the fractional order $-1<\alpha \leq 0$ and $\alpha\geq 0$. We show that the result has the same form for all values of $\alpha$. Applications can be implemented to discuss deformed potential fields resulting from fractional mass or charge densities in gravity and electrostatics problems. The result can also be applied to modify the inverse-quare law gravity as predicted by new physics.


Introduction
Fractional calculus deals with differentiation and integration to arbitrary real or complex orders. Extensive mathematical discussion of fractional calculus can be found in Refs. [1]- [4] and references therein. The techniques of fractional calculus have been applied to wide range of fields, such as physics, engineering, chemistry, biology, economics, control theory, signal image processing, groundwater problems, and many others.
Physics applications of fractional calculus span a wide range of topics and problems (for a review see Refs. [5]- [11] and references therein). Generalizing fractional calculus to several variables, multidimensional space, and generalization of fractional vector calculus has been reported [12]- [19]. Also, progress has been reported on generalization of Lagrangian and Hamiltonian systems [20]- [25].
In this work we simply consider the potential field Φ(r) = k/r, where k is constant describing the strength of the field and r = x 2 + y 2 + z 2 . The potential field emerges from an inverse-square law of gravity and Coulomb electric field. We calculate the fractional integral and derivative of Φ using Cartesian coordinates and for wide range of the fractional order α.
This calculation is important for applications to gravity and electrostatics problems. For example, one can implement the techniques of fractional calculus to relate two known mass or charge distributions by a continuous deformation as discussed in Refs. [26,27]. Thus, one can study these defor- Modifications to the inverse-quare law gravity has been are argued in theories of large extra dimensions, broken supersymmetry at low energy, and string theories, while deviation has been tested in several experiments (see Refs. [28]- [30] and references therein). In classical gravity, one can think of the inverse-square force as emerging from the specific potential field Φ(r). The two views are equivalent as they are related by the integer derivative operator, the gradient, F = −∇Φ. A possible slight deviation from the inverse square force could be due to a slight modification to the integer derivative, in other words, a fractional derivative of the potential field will lead to modification to the inverse-square force. This motivates the need to consider the fractional derivative of the potential field Φ. A different approach in modifying the inverse-square law is within fractional space [31] and where gravitational field is derived from a fractional mass distribution [32].
Finally the analytical derivation of the fractional derivative of 1/r in Cartesian coordinates could shed light on the corresponding connection with the fractional derivative in other coordinate systems, such as spherical coordinates [33]. In the next section we lay out the basic definitions of fractional calculus relevant to our study. In Section 3 we calculate the fractional integral and derivative of Φ(r). Finally, in Section 4 we provide some discussion.

Fractional Calculus
For mathematical properties of fractional derivatives and integrals one can consult Refs.[1]- [5] and the references therein. In this section we lay out the notation used in the next section as we consider the Riemann-Liouville and Caputo definitions of the fractional derivative. In this work both definitions give identical results. Let f (x, y, z) to be a real analytic function in a specific domain in the Euclidian space R 3 ; f : R 3 → R. The x-partial fractional integral or derivative of order α (keeping y and z constants) is , where a is the lower limit of x. Similarly the yand z-partial fractional integral or derivatives of order α are written as b D α y f (x, y, z) and c D α z f (x, y, z), respectively. Note that α < 0 represents a fractional integral, while α > 0 represents a fractional derivative. Since f (x, y, z) is analytic then the partial fractional derivatives are assumed to The Cauchy's repeated integration formula of the nthorder integration of the function f (x, y, z) along x, keeping y and z constants, can be written as A similar formula for the nth-order integration of the function f (x, y, z) along y, keeping x and z constants, Similarly for the nth-order integration of the function f (x, y, z) along z, keeping x and y constants, Definition 2.2. The fractional integration of order α < 0 and along x, keeping y and z constants, is defined as Similarly the fractional integration along y, keeping x and z constants, is Similarly the fractional integration along z, keeping x and y constants, is where Γ(.) is the Gamma function.
Definition 2.3. The Riemann-Liouville partial fractional derivatives of the order α > 0, where n − 1 < α < n and n ∈ N , are defined as Definition 2.4. The Caputo partial fractional derivatives of order α > 0, where n − 1 < α < n and n ∈ N , are defined as The Riemann-Liouville and Caputo definitions of the fractional derivative are related [1]- [5]. In our work, we consider the lower limit a = b = c = −∞ and since all partial derivatives of Φ vanish at the lower limit, we conclude that Riemann-Liouville and Caputo definitions of the fractional derivatives are equivalent, giving rise to the same result.

Fractional integral and derivative of 1/r potential
We consider the potential field Φ(r) = k/r, where k is constant describing the strength of the field and r = x 2 + y 2 + z 2 . In deriving the fractional integral and derivative of Φ(r) we choose the lower limit of the fractional integral to be a = b = c = −∞. We will drop the constant k in our derivation and can later be inserted with its correct dimensionality according to specific applications.

Fractional integral of 1/r
We start by calculating the fractional integral along z for −1 < α ≤ 0.
According to Eq. (6) Write ρ 2 = x 2 + y 2 and let t = z − u we get Using the spherical coordinates, r and θ, where r 2 = ρ 2 + z 2 and z = r cos θ we get Dividing the integration into the two regions 0 < t < r and t > r, we expand the integrand in terms of Legendre polynomials. Integrating over t we get the final form 2n + 1 n(n + 1) − α(α + 1) P n (cos θ) . (16) The result is divergent for θ = 0 (i.e., x = y = 0 and z > 0) and convergent everywhere else. The result agrees with Ref. [27] for −1 < α < 0. For α = 0, we have 1/Γ(−α) = −α + O(α 2 ) and it is easy to check that we retrieve the original field 1/r, as all terms vanish except the first term in the series (n=0).

Fractional derivative of 1/r
In deriving the fractional derivative of Φ(r) we choose the lower limit to be −∞. Since all partial derivatives of Φ vanish at the lower limit, we conclude that Riemann-Liouville and Caputo definitions are equivalent, giving rise to the same result. We consider first 0 < α < 1, according to Eq. (9) Following the same steps in the previous subsection we conclude that Writing ∂/∂z in terms of spherical coordinates we find [−α cos θP n (cos θ) Using the known identities of the Legendre polynomials [34] sin 2 θP ′ n = n (P n−1 − cos θP n ) , (2n + 1) cos θP n = (n + 1)P n+1 + nP n−1 , and shifting the sum appropriately we reach the final result The result is identical to the fractional integral, given in Eq. (16). Thus the fractional integral and derivative of Φ = 1/r have the same form. The result is valid for 0 < θ ≤ π and 0 ≤ α ≤ 1. For α = 0 it is easy to check that we retrieve the original field 1/r, as all terms vanish except the first term (n=0). For α = 1 all terms vanish except the second term (n=1) and we retrieve the result −P 1 (cos θ)/r 2 = −z/r 3 , as expected.
Due to the spherical symmetry of the potential, one can easily conclude The above results in Eqs. (24,25,26) are valid for all values of α > 0.
Similar to the case of 0 < α < 1 we write r 2 = x 2 + y 2 + z 2 and let t = z − u we get Dividing the integration into the two regions 0 < t < r and t > r, we expand the integrand in terms of Legendre polynomials. Integrating over t we get the final form Next we write ∂ m /∂z m in terms of spherical coordinates, similar to Eq. (21) The calculation is tedious but for 1 < α < 2 we have explicitly performed the calculation and used few of the Legendre identities. We reached the same result in Eq. (24), namely 2n + 1 n(n + 1) − α(α + 1) P n (cos θ) .
For example for α = 2 all terms vanish except for n = 2, thus it is straightforward to show that ∂ 2 Φ/∂z 2 = 2P 2 (cos θ)/r 3 as expected. In general one can show that as expected, where m ∈ N .

Discussion and Conclusions
We calculated the fractional integral and derivative of the potential Φ = 1/r. We found that for all values −1 < α ≤ 0 and α ≥ 0, 2n + 1 n(n + 1) − α(α + 1) P n (cos θ) , 2n + 1 n(n + 1) − α(α + 1) P n (sin θ cos φ) , (34) 2n + 1 n(n + 1) − α(α + 1) P n (sin θ sin φ) . (35) One can implement the fractional integral and derivative of 1/r to relate two known mass distributions by a continuous deformation, as discussed in Ref. [27]. Given a mass distribution, ρ( r), the gravitational potential, Φ( r), can be determined by solving the Poisson equation where G is the gravitational constant. To illustrate this point we apply a fractional αth-order differintegral operator, with respect to the z coordinate, to both sides of Eq. (36). Taking the lower limit a = −∞, we The commutativity of the two operators ∇ 2 and −∞ D α z is maintained in our case where the lower point is taken to be a = −∞ and thus the potential Φ( r) and all its positive-integer derivatives at the lower point vanish.
Thus the fractional potential −∞ D α z Φ( r) corresponds to the fractional mass distribution −∞ D α z ρ( r). Another application of the this work is to consider possible deviations of the inverse-square law gravitational field. Modifications are argued in theories of large extra dimensions, broken supersymmetry at low energy, and string theories (see Refs. [28][29][30] and references therein). Consider the Newtonian gravitational potential Φ(r) = Gm 1 m 2 /r. We consider the limiting case α = 1 − ǫ where ǫ << 1 and choose the special case θ = π, i.e. r = z, x = y = 0. Noting that P n (cos π) = (−1) n and using the alternating harmonic series sum one can show that to a leading order of ǫ where γ is the Euler number and λ is a introduced for dimensionality.
Similarly, we can show that The values of λ and ǫ modify the Newtonian gravitational field and thus are restricted by existing experimental constraints.