2.1 Historical Introduction to Fractional Derivatives

[17.2.1] Already at the beginning of calculus one of its founding fathers, namely G.W. Leibniz, investigated fractional derivatives [73, 72]. [17.2.2] Differentiation, denoted as dα (α∈N), obeys Leibniz’ product rule

for integer α, and Leibniz was intrigued by the analogy with the binomial theorem

where he uses the notation pα⁢f instead of fα to emphasize the formal operational analogy.

[17.3.1] Moving from integer to noninteger powers α∈R Leibniz suggests that "on peut exprimer par une serie infinie une grandeur comme" dα⁢h (with h=f⁢g). [17.3.2] As his first step he tests the idea of such a generalized differential quantity dα⁢h against the rules of his calculus. [17.3.3] In his calculus the differential relation d⁢h=h⁢d⁢x implies d⁢x=d⁢h/h and d⁢h/d⁢x=h. [17.3.4] One has, therefore, also d2⁢h=h⁢d⁢x2 and generally dα⁢h=h⁢d⁢xα. [17.3.5] Regarding dα⁢h=h⁢d⁢xα with noninteger α as a fractional differential relation subject to the rules of his calculus, however, leads to a paradox. [17.3.6] Explicitly, he finds (for α=1/2)

where d⁢x=d⁢h/h was used. [17.3.7] Many decades had to pass before Leibniz’ paradox was fully resolved.

[page 18, §1]

[18.1.1] Derivatives of noninteger (fractional) order motivated Euler to introduce the Gamma function [25]. [18.1.2] Euler knew that he needed to generalize (or interpolate, as he calls it) the product 1⋅2⋅…⋅n=n! to noninteger values of n, and he proposed an integral

for this purpose. [18.1.3] In §27-29 of [25] he immediately applies this formula to partially resolve Leibniz’ paradox, and in §28 he gives the basic fractional derivative (reproduced here in modern notation with Γ⁢n+1=n!)

valid for integer and for noninteger α,β.

[18.2.1] Generalizing eq. (2.5) to all functions that can be expanded into a power series might seem a natural step, but this "natural" definition of fractional derivatives does not really resolve Leibniz’ paradox. [18.2.2] Leibniz had implicitly assumed the rule

by demanding dα⁢h=h⁢d⁢xα for integer α. [18.2.3] One might therefore take eq. (2.6) instead of eq. (2.5) as an equally "natural" starting point (this was later done by Liouville in [76, p.3, eq. (1)]), and define fractional derivatives as

for functions representable as exponential series f⁢x∼∑kck⁢exp⁡λk⁢x. [18.2.4] Regarding the integral (a Laplace integral)

as a sum of exponentials, Liouville [76, p.7] then applied eq. (2.6) inside the integral to find

[page 19, §0]    where the last equality follows by substituting y⁢x=z in the integral. [19.0.1] If this equation is formally generalized to -β, disregarding existence of the integral, one finds

a formula similar to, but different from eq. (2.5). [19.0.2] Although eq. (2.10) agrees with eq. (2.5) for integer α it differs for noninteger α. [19.0.3] More precisely, if α=1/2 and β=-1/2, then

revealing again an inconsistency between eq. (2.5) and eq. (2.10) (resp. (2.9)).

[19.1.1] Another way to see this inconsistency is to expand the exponential function into a power series, and to apply Euler’s rule, eq. (2.5), to it. [19.1.2] One finds (with obvious notation)

and this shows that Euler’s rule (2.5) is inconsistent with the Leibniz/Liouville rule (2.6). [19.1.3] Similarly, Liouville found inconsistencies [75, p.95/96] when calculating the fractional derivative of exp⁡λ⁢x+exp⁡-λ⁢x based on the definition (2.7).

[19.2.1] A resolution of Leibniz’ paradox emerges when eq. (2.5) and (2.6) are compared for α=-1, and interpreted as integrals. [19.2.2] Such an interpretation was already suggested by Leibniz himself [73]. [19.2.3] More specifically, one has

showing that Euler’s fractional derivatives on the right hand side differs from Liouville’s and Leibniz’ idea on the left. [19.2.4] Similarly, eq. (2.5) corresponds to

[19.2.5] On the other hand, eq. (2.9) corresponds to

[page 20, §0]    [20.0.1] This shows that Euler’s and Liouville’s definitions differ with respect to their limits of integration.

[20.1.1] It has already been mentioned that Liouville defined fractional derivatives using eq. (2.7) (see [76, p.3, eq.(1)]) as

for functions representable as a sum of exponentials

[20.1.2] Liouville seems not to have recognized the necessity of limits of integration. [20.1.3] From his definition (2.7) he derives numerous integral and series representations. [20.1.4] In particular, he finds the fractional integral of order α>0 as

(see formula [A] on page 8 of [76, p.8]). [20.1.5] Liouville then gives formula [B] for fractional differentiation on page 10 of [76] as

where n-1<α<n. [20.1.6] Liouville restricts the discussion to functions represented by exponential series with λk>0 so that f⁢-∞=0. [20.1.7] Liouville also expands the coefficients λkα in (2.7) into binomial series

and inserts the expansion into his defintion (2.7) to arrive at formulae that contain the representation of integer order derivatives as limits of difference quotients (see [75, p.106ff]). [20.1.8] The results may be written as

[page 21, §0]    where the binomial coefficient αm is Γ⁢α-1⁢Γ⁢m-1/Γ⁢α+m-1. [21.0.1] Later, this idea was taken up by Grünwald [34], who defined fractional derivatives as limits of generalized difference quotients.

[21.1.1] Fourier[29] suggested to define fractional derivatives by generalizing the formula for trigonometric functions,

from α∈N to α∈R. [21.1.2] Again, this is not unique because the generalization

is also possible.

[21.2.1] Grünwald wanted to free the definition of fractional derivatives from a special form of the function. [21.2.2] He emphasized that fractional derivatives are integroderivatives, and established for the first time general fractional derivative operators. [21.2.3] His calculus is based on limits of difference quotients. [21.2.4] He studies the difference quotients [34, p.444]

with n=x-u/h and calls

the α-th differential quotient taken over the straight line from u to x [34, p.452]. [21.2.5] The title of his work emphasizes the need to introduce limits of integration into the concept of differentiation. [21.2.6] His ideas were soon elaborated upon by Letnikov (see [99])and applied to differential equations by Most [89].

[21.3.1] Riemann, like Grünwald, attempts to define fractional differentiation for general classes of functions. [21.3.2] Riemann defines the n-th differential quotient of a function f⁢x as the coeffcient of hn in the expansion of f⁢x+h into integer

[page 22, §0]    powers of h [96, p.354]. [22.0.1] He then generalizes this definition to noninteger powers, and demands that

holds for n∈N,α∈R. [22.0.2] The factor cn+α is determined such that ∂β⁡∂γ⁡f=∂β+γ⁡f holds, and found to be 1/Γ⁢n+α+1. [22.0.3] Riemann then derives the integral representation [96, p.363] for negative α

where k,Kn are finite constants. [22.0.4] He then extends the result to nonnegative α by writing "für einen Werth von α aber, der ≥0 ist, bezeichnet ∂α⁡f dasjenige, was aus ∂α-m⁡f (wo m>α) durch m-malige Differentiation nach x hervorgeht,…" [96, p.341]. [22.0.5] The combination of Liouville’s and Grünwald’s pioneering work with this idea has become the definition of the Riemann-Liouville fractional derivatives (see Section 2.2.2.1 below).