Mathematical Physics Vol 1

8.1 Brief History of Fractional Calculus

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in potential theory. Liouville began his theoretical development using the well-known result for derivatives of integer order n D n x e ax = a n e ax . (8.8) Expression (8.8) can rather easily be formally generalized to the case of non-integer values of n , thus obtaining D α x e ax = a α e ax (8.9) By means of Fourier expansion, a wide family of functions can be composed as a superposition of complex exponentials. f ( x )= ∞ ∑ n = 0 c n exp ( a n x ) , Rea n > 0 (8.10) Again, by invoking linearity of the fractional derivative, Liouville proposed the following expression for evaluating the derivative of order α

∞ ∑ n = 0

D α

α n e

a n x .

x f ( x )=

c n a

(8.11)

Previous (8.11) is known as the Liouville’s first formula for a fractional derivative,[15,16]. However, this formula cannot be seen as a general definition of fractional derivative for the same reason Lacroix formula could not: because of its relatively narrow scope. In order to overcome this, Liouville labored to produce a second definition. He started with a definite integral (closely related to the gamma function):

∞ Z 0

u β − 1 e − xu du

, β > 0 , x > 0 .

I =

(8.12)

and derived what is now referred to as the second Liouville’s formula

α Γ ( α + β ) Γ ( β )

D α

β

x − α − β

, β > 0

x x −

=( − 1 )

(8.13)

None of previous definitions were found to be suitable for a general definition of a fractional derivative. In the consequent years, a number of similar formulas emerged. Greer [17], for example, derived formulas for the fractional derivatives of trigonometric functions using (8.9) in the form: D α x e iax = a α cos πα 2 + i sin πα 2 ( cos ax + i sin ax ) (8.14) Joseph Fourier [18] obtained the following integral representations for f ( x ) and its derivatives

+ ∞ Z

+ ∞ Z

1 2 π

D n

t n cos [ t ( x − ξ )+ n π / 2 ] dt ,

f ( ξ ) d ξ

x f ( x )=

(8.15)

− ∞

− ∞

By formally replacing integer n by an arbitrary real quantity α he obtained

+ ∞ Z

+ ∞ Z

1 2 π

D α

t α cos

f ( ξ ) d ξ

[ t ( x − ξ )+ απ / 2 ] dt .

x f ( x )=

(8.16)

− ∞

− ∞

The definitions of Liouville and Lacroix are not equivalent, which led some critics to conclude that one must be "correct" and the other "wrong" [19]. De Morgan, however, wrote [20] that: " Both these systems, then, may very possibly be parts of a more general system ."