Rose-Hulman Undergraduate Mathematics Journal Volume 6 Issue 1 Article 6 Fractional Calculus and the Taylor-Riemann Series Joakim Munkhammar Uppsala University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Munkhammar, Joakim (2005) "Fractional Calculus and the Taylor-Riemann Series," Rose-Hulman Undergraduate Mathematics Journal: Vol. 6 : Iss. 1 , Article 6. Available at: https://scholar.rose-hulman.edu/rhumj/vol6/iss1/6 FRACTIONAL CALCULUS AND THE TAYLOR-RIEMANN SERIES J.D.MUNKHAMMAR ABSTRACT. In this paper we give some background theory on the con- cept of fractional calculus, in particular the Riemann-Liouville operators. We then investigate the Taylor-Riemann series using Osler’s theorem and obtain certain double infinite series expansions of some elementary func- tions. In the process of this we give a proof of the convergence of an alternative form of Heaviside’s series. A Semi-Taylor series is introduced as the special case of the Taylor-Riemann series when α = 1/2, and some of its relations to special functions are obtained via certain generating functions arising in complex fractional calculus. 1. INTRODUCTION 1.1. Riemann-Liouville operator. The concept of non-integral order of in- tegration can be traced back the to the genesis of differential calculus itself: the philosopher and creator of modern calculus G.W.Leibniz made some re- marks on the the meaning and possibility of fractional derivative of order 1/2 in the late 17:th century. However a rigorous investigation was first car- ried out by Liouville in a series of papers from 1832-1837, where he defined the first outcast of an operator of fractional integration. Later investiga- tions and further developments by among others Riemann led to the con- struction of the integral-based Riemann-Liouville fractional integral opera- tor, which has been a valuable cornerstone in fractional calculus ever since. Prior to Liouville and Riemann, Euler took the first step in the study of frac- tional integration when he studied the simple case of fractional integrals of monomials of arbitrary real order in the heuristic fashion of the time; it has been said to have lead him to construct the Gamma function for frac- tional powers of the factorial [2, p. 243]. An early attempt by Liouville was later purified by the Swedish mathematician Holmgren [13], who in 1865 made important contributions to the growing study of fractional calculus. But it was Riemann [5] who reconstructed it to fit Abel’s integral equa- tion, and thus made it vastly more useful. Today there exist many differ- ent forms of fractional integral operators, ranging from divided-difference types to infinite-sum types [1, p. xxxi], but the Riemann-Liouville Operator is still the most frequently used when fractional integration is performed. There are many interesting applications of fractional calculus, for exam- ple in physics it is used to model anomalous diffusion and in Hamiltonian Date: March 30,2005. 1 2 J.D.MUNKHAMMAR chaos fractional partial differential equations can be used [4]. Other ap- plications to physics involve fractional mechanics and fractional oscillators [12]. Applications of fractional calculus in general also appear in specula- tive option valuation in finance and are related to so called heavy tails in electrical engineering [4]. Riemann’s modified form of Liouville’s fractional integral operator is a direct generalization of Cauchy’s formula for an n-fold integral [1, p. 33]: x x1 xn−1 1 x f(t) (1) dx dx ... f(x )dx = dt. 1 2 n n (n 1)! (x t)1 n Za Za Za − Za − − By n-fold here means that the integration is deployed n-times. As an ex- ample let f(x) = x, n = 3 and a = 0 (To remove residue-terms) then (1) becomes x x1 x2 1 x t (2) x dx dx dx = dt, 3 3 2 1 2! (x t) 2 Z0 Z0 Z0 Z0 − − and by integration one gets 1 x t x4 (3) dt = , 2! (x t) 2 4! Z0 − − which gives us the 3-fold integral of f(x) = x which equals the LHS of (2). Since (n 1)! = Γ(n), Riemann realized that the RHS of (1) might − have meaning even when n takes non-integer values. Thus perhaps it was natural to define fractional integration as follows. Definition 1. If f(x) C([a, b]) and a<x<b then ∈ 1 x f(t) (4) Iα f(x) := dt a+ Γ(α) (x t)1 α Za − − where α ] , [, is called the Riemann-Liouville fractional integral of order ∈ − ∞ ∞ α. In the same fashion for α ]0, 1[ we let ∈ 1 d x f(t) (5) Dα f(x) := dt, a+ Γ(1 α) dx (x t)α − Za − which is called the Riemann-Liouville fractional derivative of order α. (It follows from our discussion below Definition 2 that if 0 <α< 1 then Dα f(x) exists for all f C1([a, b]) and all x ]a, b].) a+ ∈ ∈ These operators are called the Riemann-Liouville fractional integral op- erators, or simply R-L operators. As an example of fractional integration and differentiation one can take α = 1/2 which is called the semi-integral if used in (4) and semi-derivative if used in (5). If taking f(x)= x and letting α = 1/2 in (4) then one obtains: 1 x t (6) I1/2x = dt. 0 Γ(1/2) (x t)1/2 Z0 − FRACTIONAL CALCULUS AND THE TAYLOR-RIEMANN SERIES 3 The integral may be found in [1], hence the result will be: 1 4 (7) I1/2x = x3/2 = x3/2. 0 Γ(5/2) 3√π 3√π Since Γ(5/2) = 4 (See section 2.1). The fractional derivative for f(x)= x and α = 1/2 will by (5) become: 1 x t d 2 (8) D1/2x = dt = I1/2x = x1/2, 0 Γ(1 1/2) (x t)1 1/2 dx 0 √π − Z0 − − since Γ(3/2) = √π/2 (Additional fractional integrals may be found in Table 1). The connection between the Riemann-Liouville fractional integral and derivative can, as Riemann realized, be traced back to the solvability of Abel’s integral equation for any α ]0, 1[ ∈ 1 x φ(t) (9) f(x)= dt ,x> 0. Γ(α) (x t)1 α Za − − Formally equation (9) can be solved by changing x to t and t to s respec- tively, further by multiplying both sides of the equation by (x t) α and − − integrating we get x dt t φ(s)ds x f(t)dt (10) = Γ(α) . (x t)α (t s)1 α (x t)α Za − Za − − Za − Interchanging the order of integration in the left hand side by Fubini’s the- orem we obtain x x dt x f(t)dt (11) φ(s)ds = Γ(α) . (x t)α(t s)1 α (x t)α Za Zs − − − Za − The inner integral is easily evaluated after the change of variable t = s + τ(x s) and use of the formulae of the Beta-function (see section 2.1): − (12) x 1 α α 1 α 1 α (x t)− (t s) − dt = τ − (1 τ)− dτ = B(α, 1 α)=Γ(α)Γ(1 α). − − − − − Za Z0 Therefore we get x 1 x f(t)dt (13) φ(s)ds = . Γ(1 α) (x t)α Za − Za − Hence after differentiation we have 1 d x f(t)dt (14) φ(x)= . Γ(1 α) dx (x t)α − Za − Thus if (9) has a solution it is necessarily given by (14) for any α ]0, 1[. One ∈ observes that (9) is in a sense the α-order integral and the inversion (14) is the α-order derivative. 4 J.D.MUNKHAMMAR A very useful fact about the R-L operators is that they satisfy the follow- ing important semi-group property of fractional integrals. Theorem 1. For any f C([a, b]) the Riemann-Liouville fractional integral ∈ satisfies α β α+β (15) Ia+Ia+f(x)= Ia+ f(x) for α> 0,β > 0. Proof. The proof is rather direct, we have by definition: 1 x dt t f(u) (16) Iα Iβ f(x)= du, a+ a+ Γ(α)Γ(β) (x t)1 α (t u)1 β Za − − Za − − and since f(x) C([a, b]) we can by Fubini’s theorem interchange order of ∈ integration and by setting t = u + s(x u) we obtain − B(α,β) x f(u) (17) Iα Iβ f(x)= du = Iα+βf(x). a+ a+ Γ(α)Γ(β) (x u)1 α β a+ Za − − − ¤ The Riemann-Liouville fractional operators may in many cases be ex- tended to hold for a larger set of α, and a rather technical detail is that we denote α = [α] + α , where [α] denotes the integer part of α, and α { } { } denotes the remainder. This notation is used for convenience, observe the following definition. Definition 2. If α> 0 is not an integer then we define [α] [α]+1 α d α d 1 α (18) D f = D{ }f = I −{ }f, a+ dx[α] a+ dx[α]+1 a+ thus 1 dn x f(t)dt (19) Dα f(x)= , a+ Γ(n α) dxn (x t)α n+1 − Za − − for any f C[α]+1([a, b]) if n = [α] + 1. If on the other hand α < 0 then the ∈ notation α α (20) Da+f = Ia−+ f, may be used as definition. Clearly, if α < 0 then the fractional derivative Dα f(x) exists for all f a+ ∈ C([a, b]) and all x [a, b].