Some Notes of the Fractional Integral and Derivatives and Its Applications
In this paper, the definitions of the fractional derivative and integrals are given by the neutrix limit. Not only is it consistent with the classical results but also the representations of the fractional derivative and integrals are obtained for , , , and , where is the analytic function.
Fractional calculus has been used to model physics and engineering processes widely since standard mathematical models of integer-order derivatives, including nonlinear models. In fact, in many fields such as mechanics, electronics, chemistry, biology, economics, notably control theory, and signal and image processing, fractional calculus has been playing more and more important roles in recent years [1–5]. There are several definitions for fractional derivatives and integrals.
The Riemann–Liouville fractional integral and derivatives  are defined as.andfor and , respectively. The Grünwald–Letnikov fractional derivative  is defined aswhere is the integer part of . It has been proved that this definition is equivalent towhere . The Caputo fractional derivative, from , is defined aswhere . The fractional derivative defined above is called the left fractional integral and derivative. It is completely analogous to introducing the corresponding right fractional derivative. For example, the right Riemann–Liouville fractional derivatives is defined as
For the fractional integral and derivative of the power function , there areandfor and . When , (7) and (8) are clearly not valid. However,where the symbol represents the Pochhammer symbol, i.e.,and it requires a proper definition of the fractional derivative of . To remedy this problem, the following definition is given by Mauro Bologna :
Therefore, the traditional definition of fractional derivatives for is unreasonable. In this paper, our goal is to give a reasonable definition and representations of fractional derivatives for , , , , where is an analytic function. The properties of the fractional derivatives of the correlation function and the solution of the correlation fractional differential equation are discussed. For simplicity, we take.
Definition 1. 1 For complex number , letorandFor convenience of writing, let .
Remark 1. If , then .
The so-called neutrix limit is defined as follows.
Definition 2. Iffor , thenFor example, byfor , we haveBy exchanging the order of integration and summing, we haveFrom Definitions 2 and (23), we obtain thatfor andfor .
It should be pointed out that we can consider the fractional order derivatives of based on definitions (14)–(18), where is the analytic function and , and present the following two examples to illustrate our conclusions.
Lemma 1. For Beta function , there isfor where andfor .
Lemma 2. For the complex number and , there iswhere . In particular,
Lemma 3. For , one haswhere .Here, is the digamma function defined byand is the Hurwitz zeta function defined bywhere is the Riemann zeta function.
Lemma 4. For complex number , there is
Lemma 5. For complex number , there is.for and
Lemma 6. For , there areand
Lemma 7. For the complex number , there is
The numerical calculation shows that (40) is always true, but its theory is a little harder to prove.
3. Fractional Derivatives of and
Through the analytical continuation of , we see that (7) and (8) still hold for and . In this section, we shall consider the fractional order derivatives of and for all complex and and expect to get similar results according to the definition of neutrix limit. Our main results read as follows.
Corollary 1. (1)For the complex , there is .(2).
Proof. (1)For , Remark 1. indicates that corollary naturally holds. For , , , we have Due to for we have by Definition 2. which implies that Hence, the first part of the corollary holds.(2)From (41) and we have Then, Therefore, the second part of the corollary holds.
Theorem 2. For complex and , there isorfor andor
Proof. By (21) and exchanging the order of integration and then summing, we haveFrom (36), we haveBy (16), (35), (53), (54), Lemma 3, and Definition 2, we getwhich implies that (49) and (50) hold.
It’s worth noting that if we use the derivative method with parameter, thenwhich is exactly the same as the derivation above.
By (16) and (49), we haveIn fact, can be obtained from (29) and (57), but the expression is more complex. Here, we still use the following derivative method with parameter.so, we get (51) and (52).
Corollary 2. For the complex , there is.
Proof. For , by similar to the proof of Corollary 1, we find that (45) still holds. Hence, .
By (29), we getMoreover, from (49), we haveSubstituting (61) into (60), we getBy (40) and (62), we haveThis completes the proof of the corollary.
Theorem 3. For the complex and , there isfor andfor .