[page 17, §1]
[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 (), obeys Leibniz’ product rule
(2.1) |
for integer , and Leibniz was intrigued by the analogy with the binomial theorem
(2.2) |
where he uses the notation instead of to emphasize the formal operational analogy.
[17.3.1] Moving from integer to noninteger powers Leibniz suggests that "on peut exprimer par une serie infinie une grandeur comme" (with ). [17.3.2] As his first step he tests the idea of such a generalized differential quantity against the rules of his calculus. [17.3.3] In his calculus the differential relation implies and . [17.3.4] One has, therefore, also and generally . [17.3.5] Regarding with noninteger as a fractional differential relation subject to the rules of his calculus, however, leads to a paradox. [17.3.6] Explicitly, he finds (for )
(2.3) |
where 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 to noninteger values of , and he proposed an integral
(2.4) |
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 )
(2.5) |
valid for integer and for noninteger .
[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
(2.6) |
by demanding for integer . [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
(2.7) |
for functions representable as exponential series . [18.2.4] Regarding the integral (a Laplace integral)
(2.8) |
as a sum of exponentials, Liouville [76, p.7] then applied eq. (2.6) inside the integral to find
(2.9) |
[page 19, §0] where the last equality follows by substituting in the integral. [19.0.1] If this equation is formally generalized to , disregarding existence of the integral, one finds
(2.10) |
a formula similar to, but different from eq. (2.5). [19.0.2] Although eq. (2.10) agrees with eq. (2.5) for integer it differs for noninteger . [19.0.3] More precisely, if and , then
(2.11) |
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)
(2.12) |
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 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 , and interpreted as integrals. [19.2.2] Such an interpretation was already suggested by Leibniz himself [73]. [19.2.3] More specifically, one has
(2.13) |
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
(2.14) |
[19.2.5] On the other hand, eq. (2.9) corresponds to
(2.15) |
[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
(2.7) |
for functions representable as a sum of exponentials
(2.16) |
[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 as
(2.17) |
(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
(2.18) |
where . [20.1.6] Liouville restricts the discussion to functions represented by exponential series with so that . [20.1.7] Liouville also expands the coefficients in (2.7) into binomial series
(2.19a) | |||
(2.19b) |
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
(2.20a) | |||
(2.20b) |
[page 21, §0] where the binomial coefficient is . [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,
(2.21) |
from to . [21.1.2] Again, this is not unique because the generalization
(2.22) |
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]
(2.23) |
with and calls
(2.24) |
the -th differential quotient taken over the straight line from to [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 -th differential quotient of a function as the coeffcient of in the expansion of into integer
[page 22, §0] powers of [96, p.354]. [22.0.1] He then generalizes this definition to noninteger powers, and demands that
(2.25) |
holds for . [22.0.2] The factor is determined such that holds, and found to be . [22.0.3] Riemann then derives the integral representation [96, p.363] for negative
(2.26) |
where are finite constants. [22.0.4] He then extends the result to nonnegative by writing "für einen Werth von aber, der ist, bezeichnet dasjenige, was aus (wo ) durch -malige Differentiation nach 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).