Asymptotic Expansions of Fractional Derivatives and Their Applications

We compare the Riemann–Liouville fractional integral (fI) of a function f(z) with the Liouville fI of the same function and show that there are cases in which the asymptotic expansion of the former is obtained from those of the latter and the difference of the two fIs. When this happens, this fact occurs also for the fractional derivative (fD). This method is applied to the derivation of the asymptotic expansion of the confluent hypergeometric function, which is a solution of Kummer’s differential equation. In the present paper, the solutions of the equation in the forms of the Riemann–Liouville fI or fD and the Liouville fI or fD are obtained by using the method, which Nishimoto used in solving the hypergeometric differential equation in terms of the Liouville fD.


Introduction
We study the asymptotic expansion of the Riemann-Liouville fractional integral (fI) and fractional derivative (fD) of a function f (z), by using their relations with the corresponding Liouville fI and fD, respectively.We then present a method of deriving the asymptotic expansion of a function, when this is expressed by the Riemann-Liouville fI or fD of a function, which is analytic in a domain in the complex plane.As an example, we take up the confluent hypergeometric function.
Definition 1.Let c ∈ C, z ∈ C, f (ζ) ∈ L 1 (P (c, z)) and f (ζ) be continuous in a neighborhood of ζ = z.Then, the Riemann-Liouville fI of order λ ∈ + C is defined by: Here, P (c, z) is the path of integration from c to z, f (ζ) ∈ L 1 (P (c, z)) denotes that the function f (ζ) is integrable on P (c, z) and C, R and Z represent the sets of all complex numbers, of all real numbers and of all integers, respectively.We also use notations + C := {z ∈ C|Re z > 0}, Z >a := {n ∈ Z|n > a}, Z <b := {n ∈ Z|n < b}, Z [a,b] := {n ∈ Z|a ≤ n ≤ b} for a ∈ Z and b ∈ Z and R >0 := {x ∈ R|x > 0}.
In [1], the φ-dependent (φ-dept) Liouville fI : L D −λ φ f (z) of order λ ∈ + C is defined for φ ∈ R. It was mentioned that it is equal to (1) if c is chosen to be z + ∞ • e iφ , in [3].We now consider the paths of integration P φ (z) and P φ (c), which are from z to z + ∞ • e iφ and from c to c + ∞ • e iφ , respectively, as shown in Figure 1.Here, we consider the cases in which f (z) satisfies one of the following conditions.
Condition A. (a) f (ζ) is analytic on neighborhoods of the paths P (c, z], P φ [z] and P φ (c), and (b) it is analytic also in the region enclosed by them.If the φ-dept Liouville fI exists, the difference of the Riemann-Liouville and φ-dept Liouville fIs is expressed by the path integral along the path P φ (c).When the asymptotic expansions of the φ-dept Liouville fI and the last path integral are given, the asymptotic expansion of the Riemann-Liouville fI is obtained.
In [1], the fD corresponding to an fI is defined in the form of a contour integral, for a function f (z), which is analytic on a neighborhood of the path of integration.They are analytic continuations of the corresponding fI.As a consequence, the relation between the Riemann-Liouville fI and the corresponding φ-dept Liouville fI is analytically continued to the relation between the corresponding fDs.It follows that the same recipe is useful in obtaining the asymptotic expansion of the Riemann-Liouville fD.
In Section 2, we first recall the expressions of the Riemann-Liouville fD, which are expressed by a contour integral.As examples of functions that are expressed by the Riemann-Liouville fD, we consider the incomplete gamma function γ(λ, z) and the confluent hypergeometric function 1 F 1 (a; b; z), in Sections 2.3 and 3, respectively.
In Section 4, we recall the expressions of the φ-dept Liouville fD, which are expressed by a contour integral.We then present the φ-dept Liouville fD of the functions of which the Riemann-Liouville fD is studied, in Sections 4.4 and 5.
We show the list of fI and fD, which we use in the present paper, with the places where they are defined.
Table 1.List of the fractional integral (fI) and fractional derivative (fD) and the places of their definitions.

Riemann-Liouville fI and fD
φ-dept Liouville fI and fD In Section 6, we deform the path or contour of integration, which appears in the Riemann-Liouville fD, and show that the Riemann-Liouville fD of a function f (z) is expressed as a sum of the φ-dept Liouville fD of f (z) and a path or contour integral of f (z).By writing the asymptotic expansions of the φ-dept Liouville fD, as well as of the path or contour integral that appears, we obtain the asymptotic expansions of the Riemann-Liouville fD under consideration.We confirm that we can obtain the asymptotic expansions of γ(λ, z) and 1 F 1 (a; b; z), by this procedure in Sections 6.1∼7.
The function 1 F 1 (a; b; z) treated in Sections 3 and 7 is a solution of Kummer's differential equation (DE), and is expressed by a Riemann-Liouville fD.In Sections 5 and 7, we treat Kummer's function U (a, b, z), which is another solution of Kummer's DE and is expressed by φ-dept Liouville fD.In Sections 3.1 and 5, we show that the expressions of these functions in the form of fD are obtained by the method, which Nishimoto [4] (Chapter 5, Section 2) used in obtaining the solution of the hypergeometric DE, that is the hypergeometric function, in the form of φ-dept Liouville fI.Concluding remarks are given in Section 8.

Riemann-Liouville fI and Its Analytic Continuations
when the right-hand side (rhs) exists, where m = Re ν + 1, and Here, x for x ∈ R denotes the greatest integer not exceeding x.
We use the following index law and Leibniz's rule, in Section 3.2.Then, C D ν c f (z) is defined by: for ν ∈ C\Z <0 where the contour of integration is the Cauchy contour C(c, z + ) shown in Figure 2a, which starts from c, encircles the point z counterclockwise and goes back to c, without crossing the path , where ν ∈ C\Z.By comparing ( 7) and ( 1), we confirm the following lemma with the aid of Lemmas 6 and 4.
When n ∈ Z >−1 , (5) shows that C D n 0 e az = d n dz n e az = a n e az .

Confluent Hypergeometric Function in Terms of Riemann-Liouville fD
For a ∈ C, z ∈ C and b ∈ C\Z <1 , the confluent hypergeometric series 1 F 1 (a; b; z) is defined by: where when Re a > 0 and Re (b − a) > 0, in [11] (Section 13.2.1).This function is a solution of Kummer's DE for a ∈ C, b ∈ C and z ∈ C: This DE has also another solution given by: see [11] (Section 13.1.13).

Solution of Kummer's DE in Terms of Riemann-Liouville fD
The function 1 F 1 (a; b; z) is known to be expressed in the form of ( 13) given below, in [8].We now obtain the solutions of (11) expressed in terms of the Riemann-Liouville fD, by using the method that Nishimoto [4] adopted in deriving the solution of the hypergeometric DE, that is the hypergeometric function, in the form of the Liouville fD.
Proofs of the following two lemmas are presented in the following section.
Lemma 9.There exist the following solutions of (11): where: The values of a l , b l , δ l and their linear combinations are given in Table 2.
Table 2. Values of a l , b l , δ l and their linear combinations. .

Proofs of Lemmas 9 and 10
In place of (11), we now consider the solution of the equation: where δ ∈ C. In the proof of Lemma 9 given below, u l (z) in ( 13) are the solutions of (18) for a = a l , b = b l and δ = δ l , which are listed in Table 2.
The following lemma is obtained by using the method of Nishimoto [4] mentioned above.
Lemma 11.Let Re a < −1 and Re (a − b) ≥ 0.Then, a solution of ( 18) is given by: Proof We assume that a solution of ( 18) is expressed as Substituting this w(z) in (18) and using Leibniz's rule given in Lemma 3, we obtain: Putting λ = 1 − a and hence assuming Re a < −1, and applying R D λ−2 0 to (20), we obtain: with the aid of Lemma 2. This equation requires that: and z • v(z) = 0 when z = 0. Now, we obtain v(z) = z a−b e δ•z .Thus, we obtain (19).

Proof of Lemma 9
The formula for w 1 (z) in ( 13) follows from Lemma 11 with the aid of Lemmas 5, 6 and 7. We next give a derivation of the formulas for w 2 (z) and w 3 (z) in (13).For l = 2 and l = 3, we put w(z) = p l (z)u l (z) in (11) and obtain (18) with w, a, b and δ replaced by u l , a l , b l and δ l , respectively.We next give a derivation of the formula for w 4 (z) in (13).We put w(z) = p 4 (z)u 4 (z) = p 3 (z)p 2 (z)u 4 (z) in (11), and then, we obtain (18) with w, a, b and δ replaced by Proof of Lemma 10 We put ) .By using Lemmas 7 and 6 and the last member of (1), ( 15) is expressed as: when 1 + Re (a l − b l ) > 0 and 1 − Re a l > 0. This gives: By using (24) in ( 13), we obtain ( 16)∼( 17).

Definitions of φ-dept Liouville fI
Let z ∈ C and φ ∈ R. We denote the half line {z + te iφ | 0 < t < ∞}, as shown in Figure 1b, by P φ (z).When f (z + te iφ ) is locally integrable as a function of t in the interval (0, ∞), we denote this by f (z) ∈ L 1 loc (P φ (z)).
)|dt converges for s > s 1 and diverges for s < s 1 .We then call s 1 the abscissa of convergence, and denote it by We note that there exists such a series Then, we define two types of fI, W D −λ φ f (z) and L D −λ φ f (z), by: We call L D −λ φ f (z) the φ-dept Liouville fI of f (z).

Definitions of φ-dept Liouville fD
Let z ∈ C and φ ∈ R. When f (z + te iφ ) is infinitely differentiable as a function of t in the interval (0, ∞), we denote this by f (z) ∈ C ∞ (P φ (z)).
, and m = Re ν + 1.We then define fD, W D ν φ f (z), by: where . Formula (28) applies when Re ν − 1 > s 1 [f ] and the rhs exists.When W D ν φ f (z) exists, L D ν φ f (z) also exists and is given by: We call L D ν φ f (z) the φ-dept Liouville fD of f (z).

Analytic Continuations of φ-dept Liouville fI
In [1,4,12,13], analytic continuations of φ-dept Liouville fI via contour integrals are discussed.The analytic continuation via Hankel's contour C φ (z), which is shown in Figure 3, is defined as follows.The following lemmas follow from the arguments there.
We now present the index law and Leibniz's rule corresponding to Lemmas 2 and 3.
Proof We see that both sides are equal to: In the following sections, we use H D ν φ f (z) for the Liouville fD.In particular, by using (25).The proof of the second equation of ( 34) is finished by using (26) and Lemma 14.
We present the following lemma, but we will not use it later.

Solution of Kummer's DE in Terms of φ-dept Liouville fD
1 F 1 (a; b; z) is a solution of Kummer's DE (11), as stated in Section 3. Kummer's function U (a, b, z) is another solution of (11), which has the integral representation given by: when Re a > 0 and Re z > 0 [11] (Section 13.2.5).
We now give a solution of (11), following the method adopted in Section 3.1 for the Riemann-Liouville fD.Proofs of the lemmas in this section are presented in the following section.
Corresponding to Lemma 9, we now have the following lemma.
Lemma 20.There exist the following solutions of (11): where p l (z) are given in (14), and: The values of a l , b l , δ l , and their linear combinations are given in Table 2.
Remark 3. We find a solution of Kummer's DE in [15] (Chapter VII, Section 8), where a solution is first obtained for the equation adjoint to Kummer's DE, by the method we adopted above, and then, a relation connecting the two solutions is used to give the above solution w4 (z) = U (a, b, z).

Proofs of Lemmas 20 and 21
The following lemma is obtained by using the method of Nishimoto [4], which is mentioned in Section 3.1.
Lemma 22.A solution of (18) for δ = 1 or −1 is: Proof We assume that a solution of (18) is expressed as w(z) = H D ν φ v(z) for ν ∈ C. Substituting this w(z) in (18) and using Leibniz's rule given in Lemma 17, we obtain (20) with λ, R and zero replaced by −ν, H and φ, respectively.Putting ν = a − 1 and then applying H D −ν−1 Proof of Lemma 20 Equation (40) for l = 1 follows from Lemma 22.For l = 2, 3 and 4, see the proof of Lemma 9.For φ − (1 + δ l )π/2, see the last part of the proof of Lemma 22.The factors e iπ(a l −1) are so chosen that (40) gives Lemma 21.
Proof of Lemma 21 By using Lemmas 15 and 14 and the last member of (26), ( 41) is expressed as: when −Re a l > −1.By putting η = e −iπ δ l • zt, we obtain: when −Re a l + 1 > 0 and −δ l • Re z > 0. The integral in (39) is equal to that in (46) with a, b and z replaced by 1 − a l , 2 − b l and e −iπ δ l • z, respectively, and hence, we have: By using this in (40), we obtain (42)∼(43).

Asymptotic Expansions of Riemann-Liouville fD
Comparing the third member of (26) with (1) and Figure 1, we have the following lemma.
Lemma 23.Let Condition A in Section 1 be satisfied, Comparing ( 5) and (30), Figures 2a and 4a, we have the following lemma.
We express the integral on the rhs of (49) as: where y = z − c and η = ζ − c.We now obtain the asymptotic expansion of (51) as a function of y.
We use the Taylor series given in [10] (Section 5.4): for a ∈ C, h ∈ C, µ ∈ C and n ∈ Z >0 , where: Here, C is a contour, including a and a + h, but not zero.We now choose α ∈ R >0 and β ∈ R >0 satisfying α < 1 and β < 1, and choose C, such that ξ ∈ C on C satisfies: We then choose r ∈ R >0 , such that: Remark 6.In using the Taylor series (52), we put a = z − c, h being on P φ (0) or C φ (0), which are shown in Figures 1 and 3, respectively, and hence, a+h is on P φ (z −c) or C φ (z −c).If we see Figure 4a with c and z replaced by zero and z − c, respectively, we can easily choose a contour C, as described above.
We then obtain: where: Using ( 52) with (56), we obtain the following lemma.
where: Remark 9. When φ satisfies z−c = |z−c|•e i(φ+π) , Theorems 1 and 2 are valid even when the Condition (b) in Condition A is not satisfied.Formula (68) given below is a trivial example.

fD of the Exponential Function
We now apply Theorem 1 to the function f (z) = e −az .By using (34), we obtain the following results.
Remark 10.In the above derivation of the second term on the rhs of (65), we use (52) in the integral ∞ 0 (x + s) −ν−1 e −as ds.This derivation is the one given in [10] (Section 16.3), for Whittaker's function W k,m (z).
When a = −ib, we have: Corollary 2. Let b ∈ R\{0}, x ∈ R and ν ∈ C\Z >−1 .Then, the asymptotic expansion of C D ν 0 e ibx is given by (65) with −a and z replaced by ib and x, respectively.Remark 11.Sakakibara [16] used the fact that the leading term of the asymptotic expansion of C D 1/2 0 e iγx for γ ∈ R and x 1 is √ iγe iγx .

Concluding Remarks
We presented a method of deriving the asymptotic expansion of a function, which is expressed by the Riemann-Liouville fI or fD, when these are given by contour integrals.The derivation of the formulas is done by using contour integrals, and yet, the final formulas are mostly useful for the functions of real variable.
Here, we mention [17] reviewing recent developments on fractional calculus.

2. 1 .
Definition of Riemann-Liouville fD and the Index Law of Riemann-Liouville fI Definition 2. The Riemann-Liouville fD of order ν ∈ C satisfying Re ν ≥ 0 is defined by:

Lemma 2 .Definition 3 .
Let λ ∈ + C, ν ∈ C satisfy Re ν ≤ Re λ and R D −λ c f (z) exist.Then: Let f (ζ) be analytic on a neighborhood of the path P (c, z] and be integrable on P (c, z).