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2.2 Mathematical Introduction to Fractional Derivatives

[22.1.1] The brief historical introduction has shown that fractional derivatives may be defined in numerous ways. [22.1.2] A natural and frequently used approach starts from repeated integration and extends it to fractional integrals. [22.1.3] Fractional derivatives are then defined either by continuation of fractional integrals to negative order (following Leibniz’ ideas [73]), or by integer order derivatives of fractional integrals (as suggested by Riemann [96]).

2.2.1 Fractional Integrals

2.2.1.1 Iterated Integrals

[22.2.1] Consider a locally integrable1 (This is a footnote:) 1 A function f:\mathbb{G}\to\mathbb{R} is called locally integrable if it is integrable on all compact subsets K\subset\mathbb{G} (see eq.(B.9)). real valued function f:\mathbb{G}\to\mathbb{R} whose domain of definition \mathbb{G}=[a,b]\subseteq\mathbb{R} is an interval with -\infty\leq a<b\leq\infty. [22.2.2] Integrating

[page 23, §0]    n times gives the fundamental formula

\displaystyle(\mathrm{I}_{{a+}}^{{n}}f)(x) \displaystyle=\int\limits _{a}^{x}\int\limits _{a}^{{x_{1}}}...\int\limits _{a}^{{x_{{n-1}}}}f(x_{n})\;\mathrm{d}x_{n}...\mathrm{d}x_{2}\mathrm{d}x_{1}
\displaystyle=\frac{1}{(n-1)!}\int\limits _{a}^{x}(x-y)^{{n-1}}f(y)\;\mathrm{d}y, (2.27)

where a<x<b and n\in\mathbb{N}. [23.0.1] This formula may be proved by induction. [23.0.2] It reduces n-fold integration to a single convolution integral (Faltung). [23.0.3] The subscript a+ indicates that the integration has a as its lower limit. [23.0.4] An analogous formula holds with lower limit x and upper limit a. [23.0.5] In that case the subscript a- will be used.

2.2.1.2 Riemann-Liouville Fractional Integrals

[23.1.1] Equation (2.27) for n-fold integration can be generalized to noninteger values of n using the relation (n-1)!=\prod _{{k=1}}^{{n-1}}k=\Gamma(n) where

\Gamma(z)=\int\limits _{0}^{1}(-\log x)^{{z-1}}\;\mathrm{d}x (2.28)

is Euler’s \Gamma-function defined for all z\in\mathbb{C}.

Definition 2.1

[23.2.1] Let -\infty\leq a<x<b\leq\infty. [23.2.2] The Riemann-Liouville fractional integral of order \alpha>0 with lower limit a is defined for locally integrable functions f:[a,b]\to\mathbb{R} as

(\mathrm{I}_{{a+}}^{{\alpha}}f)(x)=\frac{1}{\Gamma(\alpha)}\int\limits _{a}^{x}(x-y)^{{\alpha-1}}f(y)\;\mathrm{d}y (2.29a)
for x>a. [23.2.3] The Riemann-Liouville fractional integral of order \alpha>0 with upper limit b is defined as
(\mathrm{I}_{{b-}}^{{\alpha}}f)(x)=\frac{1}{\Gamma(\alpha)}\int\limits _{x}^{b}(y-x)^{{\alpha-1}}f(y)\;\mathrm{d}y (2.29b)

for x<b. [23.2.4] For \alpha=0

(\mathrm{I}_{{a+}}^{{0}}f)(x)=(\mathrm{I}_{{b-}}^{{0}}f)(x)=f(x) (2.30)

completes the definition. [23.2.5] The definition may be generalized to \alpha\in\mathbb{C} with \mathrm{Re}\,\alpha>0.

[page 24, §1]    [24.1.1] Formula (2.29a) appears in [96, p.363] with a>-\infty and in [76, p.8] with a=-\infty. [24.1.2] The notation is not standardized. [24.1.3] Leibniz, Lagrange and Liouville used the symbol \int^{\alpha} [73, 22, 76], Grünwald wrote \int^{\alpha}[...\mathrm{d}x^{\alpha}]^{{x=x}}_{{x=a}}, while Riemann used \partial^{{-\alpha}}_{x} [96] and Most wrote \mathrm{d}_{a}^{{-\alpha}}/\mathrm{d}x^{{-\alpha}} [89]. [24.1.4] The notation in (2.29) is that of [99, 98, 54, 52]. [24.1.5] Modern authors also use f_{\alpha} [37], \mathrm{I}^{{\alpha}} [97], {}_{a}I_{x}^{\alpha} [94], I^{\alpha}_{x} [23], {}_{a}D_{x}^{{-\alpha}} [85, 102, 91], or \mathrm{d}^{{-\alpha}}/\mathrm{d}(x-a)^{{-\alpha}} [92] instead of \mathrm{I}_{{a+}}^{{\alpha}}2 (This is a footnote:) 2 Some authors [97, 26, 92, 23, 85, 91] employ the derivative symbol D also for integrals, resp. I for derivatives, to emphasize the similarity between fractional integration and differentiation. If this is done, the choice of Riesz and Feller, namely I, seems superior in the sense that fractional derivatives, similar to integrals, are nonlocal operators, while integer derivatives are local operators..

[24.2.1] The fractional integral operators \mathrm{I}_{{a+}}^{{\alpha}},\mathrm{I}_{{b-}}^{{\alpha}} are commonly called Riemann-Liouville fractional integrals [99, 98, 94] although sometimes this name is reserved for the case a=0 [85]. [24.2.2] Their domain of definition is typically chosen as D(\mathrm{I}_{{a+}}^{{\alpha}})=L^{{1}}([a,b]) or D(\mathrm{I}_{{a+}}^{{\alpha}})=L^{{1}}_{{\mathrm{loc}}}([a,b]) [99, 98, 94]. [24.2.3] For the definition of Lebesgue spaces see the Appendix B. [24.2.4] If f\in L^{{1}}([a,b]) then (\mathrm{I}_{{a+}}^{{\alpha}}f)\in L^{{1}}([a,b]) and (\mathrm{I}_{{a+}}^{{\alpha}}f)(x) is finite for almost all x. [24.2.5] If f\in L^{{p}}([a,b]) with 1\leq p\leq\infty and \alpha>1/p then (\mathrm{I}_{{a+}}^{{\alpha}}f)(x) is finite for all x\in[a,b]. [24.2.6] Analogous statements hold for (\mathrm{I}_{{b-}}^{{\alpha}}f)(x) [98].

[24.3.1] A short table of Riemann-Liouville fractional integrals is given in Appendix A. [24.3.2] For a more extensive list of fractional integrals see [24].

2.2.1.3 Weyl Fractional Integrals

[24.4.1] Examples (2.5) and (2.6) or (A.3) and (A.5) show that Definition 2.1 is well suited for fractional integration of power series, but not for functions defined by Fourier series. [24.4.2] In fact, if f(x) is a periodic function with period 2\pi, and3 (This is a footnote:) 3The notation \sim indicates that the sum does not need to converge, and, if it converges, does not need to converge to f(x).

f(x)\sim\sum _{{k=-\infty}}^{\infty}c_{k}\mathrm{e}^{{\mathrm{i}kx}} (2.31)

then the Riemann-Liouville fractional (\mathrm{I}_{{a+}}^{{\alpha}}f) will in general not be periodic. [24.4.3] For this reason an alternative definition of fractional integrals was investigated by Weyl [124].

[24.5.1] Functions on the unit circle \mathbb{G}=\mathbb{R}/2\pi\mathbb{Z} correspond to 2\pi-periodic functions on the real line. [24.5.2] Let f(x) be periodic with period 2\pi and such that the integral of f over the interval [0,2\pi] vanishes, so that c_{0}=0 in eq. (2.31). [24.5.3] Then the integral of f is itself a periodic function, and the constant of integration can be chosen such that the integral over [0,2\pi] vanishes again. [24.5.4] Repeating the integration n times one finds using (2.6) and the integral representation

[page 25, §0]    c_{k}=(1/2\pi)\int _{{0}}^{{2\pi}}\mathrm{e}^{{-iks}}f(s)\mathrm{d}s of Fourier coefficients

\sum _{{k=-\infty}}^{\infty}c_{k}\frac{e^{{ikx}}}{(ik)^{n}}=\frac{1}{2\pi}\int\limits _{0}^{{2\pi}}f(y)\sum _{{\substack{k=-\infty\\ k\neq 0}}}^{\infty}\frac{e^{{ik(x-y)}}}{(ik)^{n}}\mathrm{d}y (2.32)

with c_{0}=0. [25.0.1] Recall the convolution formula [132, p.36]

(f*g)(t)=\frac{1}{2\pi}\int\limits _{0}^{{2\pi}}f(t-s)g(s)\mathrm{d}s=\sum _{{k=-\infty}}^{\infty}f_{k}g_{k}\mathrm{e}^{{ikt}} (2.33)

for two periodic functions f(t)\sim\sum _{{k=-\infty}}^{\infty}f_{k}\mathrm{e}^{{ikt}} and g(t)\sim\sum _{{k=-\infty}}^{\infty}g_{k}\mathrm{e}^{{ikt}}. [25.0.2] Using eq. (2.33) and generalizing (2.32) to noninteger n suggests the following definition. [99, 94].

Definition 2.2

[25.1.1] Let f\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}),1\leq p<\infty be periodic with period 2\pi and such that its integral over a period vanishes. [25.1.2] The Weyl fractional integral of order \alpha is defined as

(\mathrm{I}_{{\pm}}^{{\alpha}}f)(x)=(\Psi _{\pm}^{\alpha}*f)(x)=\frac{1}{2\pi}\int\limits _{0}^{{2\pi}}\Psi _{\pm}^{\alpha}(x-y)f(y)\mathrm{d}y, (2.34)

where

\Psi _{\pm}^{\alpha}(x)=\sum _{{\substack{k=-\infty\\ k\neq 0}}}^{\infty}\frac{e^{{ikx}}}{(\pm\mathrm{i}k)^{\alpha}} (2.35)

for 0<\alpha<1.

[25.2.1] It can be shown that the series for \Psi _{\pm}^{\alpha}(x) converges and that the Weyl definition coincides with the Riemann-Liouville definition [133]

(\mathrm{I}_{{+}}^{{\alpha}}f)(x)=\frac{1}{\Gamma(\alpha)}\int\limits _{{-\infty}}^{x}(x-y)^{{\alpha-1}}f(y)\;\mathrm{d}y, (2.36a)
respectively
(\mathrm{I}_{{-}}^{{\alpha}}f)(x)=\frac{1}{\Gamma(\alpha)}\int\limits _{{x}}^{\infty}(y-x)^{{\alpha-1}}f(y)\;\mathrm{d}y (2.36b)

for 2\pi periodic functions whose integral over a period vanishes. [25.2.2] This is eq. (2.29) with a=-\infty resp. b=\infty. [25.2.3] For this reason the Riemann-Liouville

[page 26, §0]    fractional integrals with limits \pm\infty, \mathrm{I}_{{+}}^{{\alpha}}f=\mathrm{I}_{{(-\infty)+}}^{{\alpha}}f and \mathrm{I}_{{-}}^{{\alpha}}f=\mathrm{I}_{{\infty-}}^{{\alpha}}f, are often called Weyl fractional integrals [24, 99, 85, 94].

[26.1.1] The Weyl fractional integral may be rewritten as a convolution

(\mathrm{I}_{{\pm}}^{{\alpha}}f)(x)=(K_{\pm}^{\alpha}*f)(x), (2.37)

where the convolution product for functions on \mathbb{R} is defined as4 (This is a footnote:) 4 If K,f\in L^{{1}}(\mathbb{R}) then (K*f)(t) exists for almost all t\in\mathbb{R} and f\in L^{{1}}(\mathbb{R}). [26.1.2] If K\in L^{{p}}(\mathbb{R}), f\in L^{{q}}(\mathbb{R}) with 1<p,q<\infty and 1/p+1/q=1 then K*f\in C_{0}(\mathbb{R}), the space of continuous functions vanishing at infinity.

(K*f)(x):=\int\limits _{{-\infty}}^{\infty}K(x-y)f(y)\mathrm{d}y (2.38)

and the convolution kernels are defined as

K_{\pm}^{\alpha}(x):=\Theta(\pm x)\frac{(\pm x)^{{\alpha-1}}}{\Gamma(\alpha)} (2.39)

for \alpha>0. [26.1.3] Here

\Theta(x)=\begin{cases}1&,x>0\\ 0&,x\leq 0\end{cases} (2.40)

is the Heaviside unit step function, and x^{\alpha}=\exp{\alpha\log x} with the convention that \log x is real for x>0. [26.1.4] For \alpha=0 the kernel

K_{+}^{0}(x)=K_{-}^{0}(x)=\delta(x) (2.41)

is the Dirac \delta-function defined in (C.2) in Appendix C. [26.1.5] Note that K_{\pm}^{\alpha}\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{R}) for \alpha>0.

2.2.1.4 Riesz Fractional Integrals

[26.2.1] Riemann-Liouville and Weyl fractional integrals have upper or lower limits of integration, and are sometimes called left-sided resp. right-sided integrals. [26.2.2] A more symmetric definition was advanced in [97].

Definition 2.3

[26.3.1] Let f\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{R}) be locally integrable. [26.3.2] The Riesz fractional integral or Riesz potential of order \alpha>0 is defined as the linear combination [99]

\displaystyle(\mathrm{I}^{{\alpha}}f)(x) \displaystyle=\frac{(\mathrm{I}_{{+}}^{{\alpha}}f)(x)+(\mathrm{I}_{{-}}^{{\alpha}}f)(x)}{2\cos(\alpha\pi/2)}
\displaystyle=\frac{1}{2\Gamma(\alpha)\cos(\alpha\pi/2)}\int\limits _{{-\infty}}^{{\infty}}\frac{f(y)}{|x-y|^{{1-\alpha}}}\mathrm{d}y (2.42)

[page 27, §0]    of right- and left-sided Weyl fractional integrals. [27.0.1] The conjugate Riesz potential is defined by

\displaystyle(\widetilde{\mathrm{I}^{{\alpha}}}f)(x) \displaystyle=\frac{(\mathrm{I}_{{+}}^{{\alpha}}f)(x)-(\mathrm{I}_{{-}}^{{\alpha}}f)(x)}{2\sin(\alpha\pi/2)}
\displaystyle=\frac{1}{2\Gamma(\alpha)\sin(\alpha\pi/2)}\int\limits _{{-\infty}}^{{\infty}}\frac{\operatorname{sgn}(x-y)f(y)}{|x-y|^{{1-\alpha}}}\mathrm{d}y. (2.43)

[27.0.2] Of course, \alpha\neq 2k+1,k\in\mathbb{Z} in (2.42) and \alpha\neq 2k,k\in\mathbb{Z} in (2.43). [27.0.3] The definition is again completed with

(\mathrm{I}^{{0}}f)(x)=(\widetilde{\mathrm{I}^{{0}}}f)(x)=f(x) (2.44)

for \alpha=0.

[27.1.1] Riesz fractional integration may be written as a convolution

\displaystyle(\mathrm{I}^{{\alpha}}f)(x) \displaystyle=(K^{\alpha}*f)(x) (2.45a)
\displaystyle(\widetilde{\mathrm{I}^{{\alpha}}}f)(x) \displaystyle=(\widetilde{K}^{\alpha}*f)(x) (2.45b)

with the (one-dimensional) Riesz kernels

K^{\alpha}(x)=\frac{K^{\alpha}_{+}(x)+K^{\alpha}_{-}(x)}{2\cos(\alpha\pi/2)}=\frac{|x|^{{\alpha-1}}}{2\cos(\alpha\pi/2)\Gamma(\alpha)} (2.46)

for \alpha\neq 2k+1,k\in\mathbb{Z}, and

\widetilde{K}^{\alpha}(x)=\frac{K^{\alpha}_{+}(x)-K^{\alpha}_{-}(x)}{2\sin(\alpha\pi/2)}=\frac{|x|^{{\alpha-1}}\operatorname{sgn}(x)}{2\sin(\alpha\pi/2)\Gamma(\alpha)} (2.47)

for \alpha\neq 2k,k\in\mathbb{Z}. [27.1.2] Subsequently, Feller introduced the generalized Riesz-Feller kernels [26]

K^{{\alpha,\beta}}(x)=\frac{|x|^{{\alpha-1}}\sin\left[\alpha\left(\pi/2+\beta\mathrm{sgn}\, x\right)\right]}{2\sin(\alpha\pi/2)\Gamma(\alpha)} (2.48)

with parameter \beta\in\mathbb{R}. [27.1.3] The corresponding generalized Riesz-Feller fractional integral of order \alpha and type \beta is defined as

(\mathrm{I}^{{\alpha,\beta}}f)(x)=(K^{{\alpha,\beta}}*f)(x). (2.49)

[27.1.4] This formula interpolates continuously from the Weyl integral \mathrm{I}_{{-}}^{{\alpha}}=\mathrm{I}^{{\alpha,-\pi/2}} for \beta=-\pi/2 through the Riesz integral \mathrm{I}^{{\alpha}}=\mathrm{I}^{{\alpha,0}} for \beta=0 to the Weyl integral \mathrm{I}_{{+}}^{{\alpha}}=\mathrm{I}^{{\alpha,\pi/2}} for \beta=\pi/2. [27.1.5] Due to their symmetry Riesz-Feller fractional integrals are readily generalized to higher dimensions.

[page 28, §1]

2.2.1.5 Fractional Integrals of Distributions

[28.1.1] Fractional integration can be extended to distributions using the convolution formula (2.37) above. [28.1.2] Distributions are generalized functions [105, 31]. [28.1.3] They are defined as linear functionals on a space X of conveniently chosen ‘‘test functions’’. [28.1.4] For every locally integrable function f\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{R}) there exists a distribution \mathit{F}_{f}:X\to\mathbb{C} defined by

\mathit{F}_{f}(\varphi)=\langle f,\varphi\rangle=\int\limits _{{-\infty}}^{\infty}f(x)\varphi(x)\;\mathrm{d}x, (2.50)

where \varphi\in X is test function from a suitable space X of test functions. [28.1.5] By abuse of notation one often writes f for the associated distribution \mathit{F}_{f}. [28.1.6] Distributions that correspond to functions via (2.50) are called regular distributions. [28.1.7] Examples for regular distributions are the convolution kernels K_{\pm}^{\alpha}\in L^{{1}}_{{\mathrm{loc}}}(\mathbb{R}) defined in (2.39). [28.1.8] They are locally integrable functions on \mathbb{R} when \alpha>0. [28.1.9] Distributions that are not regular are sometimes called singular. [28.1.10] An important example for a singular distribution is the Dirac \delta-function. [28.1.11] It is defined as \delta:X\to\mathbb{C}

\int\limits\delta(x)\varphi(x)\mathrm{d}x=\varphi(0) (2.51)

for every test function \varphi\in X. [28.1.12] The test function space X is usually chosen as a subspace of C^{{\infty}}(\mathbb{R}), the space of infinitely differentiable functions. [28.1.13] A brief introduction to distributions is given in Appendix C.

[28.2.1] In order to generalize (2.37) to distributions one must define the convolution of two distributions. [28.2.2] To do so one multiplies eq. (2.38) on both sides with a smooth test function \varphi\in C_{{\mathrm{c}}}^{{\infty}}(\mathbb{R}) of compact support. [28.2.3] Integrating gives

\displaystyle\langle K*f,\varphi\rangle \displaystyle=\int\limits _{{-\infty}}^{\infty}\int\limits _{{-\infty}}^{\infty}K(x-y)f(y)\varphi(x)\mathrm{d}y\mathrm{d}x
\displaystyle=\int\limits _{{-\infty}}^{\infty}\int\limits _{{-\infty}}^{\infty}K(x)f(y)\varphi(x+y)\mathrm{d}y\mathrm{d}x
\displaystyle=\langle K(x),\langle f(y),\varphi(x+y)\rangle\rangle, (2.52)

where the notation \langle f(y),\varphi(x+y)\rangle means that the functional \mathit{F}_{f} is applied to the function \varphi(x+\cdot) for fixed x. [28.2.4] Explicitly, for fixed x

\mathit{F}_{f}(\varphi _{x})=\langle f(y),\varphi _{x}(y)\rangle=\langle f(y),\varphi(x+y)\rangle=\int\limits _{{-\infty}}^{\infty}f(y)\varphi(x+y)\mathrm{d}x, (2.53)

[page 29, §0]    where \varphi _{x}(\cdot)=\varphi(x+\cdot). [29.0.1] Equation (2.52) can be used as a definition for the convolution of distributions provided that the right hand side has meaning. [29.0.2] This is not always the case as the counterexample K=f=1 shows. [29.0.3] In general the convolution product is not associative (see eq. (2.113)). [29.0.4] However, associative and commutative convolution algebras exist [21]. [29.0.5] Equation (2.52) is always meaningful when \mathrm{supp}\, K or \mathrm{supp}\, f is compact [63]. [29.0.6] Another case is when K and f have support in \mathbb{R}_{+}. [29.0.7] This will be assumed in the following.

Definition 2.4

[29.1.1] Let f be a distribution f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, f\subset\mathbb{R}_{+}. [29.1.2] Then its fractional integral is the distribution \mathrm{I}_{{0+}}^{{\alpha}}f defined as

\langle\mathrm{I}_{{0+}}^{{\alpha}}f,\varphi\rangle=\langle\mathrm{I}_{{+}}^{{\alpha}}f,\varphi\rangle=\langle K_{+}^{\alpha}*f,\varphi\rangle (2.54)

for \mathrm{Re}\,\alpha>0. [29.1.3] It has support in \mathbb{R}_{+}.

[29.2.1] If f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, f\subset\mathbb{R}_{+} then also \mathrm{I}_{{0+}}^{{\alpha}}f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\,\mathrm{I}_{{0+}}^{{\alpha}}f\subset\mathbb{R}_{+}.

2.2.1.6 Integral Transforms

[29.3.1] The Fourier transformation is defined as

\displaystyle{\mathcal{F}}\left\{ f\right\}(k) \displaystyle=\int\limits _{{-\infty}}^{\infty}\mathrm{e}^{{-\mathrm{i}kx}}f(x)\;\mathrm{d}x (2.55)

for functions f\in L^{{1}}(\mathbb{R}). [29.3.2] Then

\displaystyle{\mathcal{F}}\left\{\mathrm{I}_{{\pm}}^{{\alpha}}f\right\}(k) \displaystyle=(\pm\mathrm{i}k)^{{-\alpha}}{\mathcal{F}}\left\{ f\right\}(k) (2.56)

holds for 0<\alpha<1 by virtue of the convolution theorem. [29.3.3] The equation cannot be extended directly to \alpha\geq 1 because the Fourier integral on the left hand side may not exist. [29.3.4] Consider e.g. \alpha=1 and f\in C_{{\mathrm{c}}}^{{\infty}}(\mathbb{R}). [29.3.5] Then (\mathrm{I}_{{+}}^{{1}}f)(x)\toconst as x\to\infty and {\mathcal{F}}\left\{\mathrm{I}_{{+}}^{{1}}f\right\} does not exist [94]. [29.3.6] Equation (2.56) can be extended to all \alpha with \mathrm{Re}\,\alpha>0 for functions in the so called Lizorkin space [99, p.148] defined as the space of functions f\in\mathcal{S}{(\mathbb{R})} such that (\mathrm{D}^{m}{\mathcal{F}}\left\{ f\right\})(0)=0 for all m\in\mathbb{N}_{0}.

[29.4.1] For the Riesz potentials one has

\displaystyle{\mathcal{F}}\left\{\mathrm{I}^{{\alpha}}f\right\}(k) \displaystyle=|k|^{{-\alpha}}{\mathcal{F}}\left\{ f\right\}(k) (2.57a)
\displaystyle{\mathcal{F}}\left\{\widetilde{\mathrm{I}^{{\alpha}}}f\right\}(k) \displaystyle=(-\mathrm{i}\mathrm{sgn}\, k)|k|^{{-\alpha}}{\mathcal{F}}\left\{ f\right\}(k) (2.57b)

for functions in Lizorkin space.

[page 30, §1]    [30.1.1] The Laplace transform is defined as

{\mathcal{L}}\left\{ f\right\}(u)=\int\limits _{0}^{\infty}\mathrm{e}^{{-ux}}f(x)\;\mathrm{d}x (2.58)

for locally integrable functions f:\mathbb{R}_{+}\to\mathbb{C}. [30.1.2] Now

{\mathcal{L}}\left\{\mathrm{I}_{{0+}}^{{\alpha}}f\right\}(u)=u^{{-\alpha}}{\mathcal{L}}\left\{ f\right\}(u) (2.59)

by the convolution theorem for Laplace transforms. [30.1.3] The Laplace transform of \mathrm{I}_{{0-}}^{{\alpha}}f leads to a more complicated operator.

2.2.1.7 Fractional Integration by Parts

[30.2.1] If f(x)\in L^{{p}}([a,b]),g\in L^{{q}}([a,b]) with 1/p+1/q\leq 1+\alpha, p,q\geq 1 and p\neq 1, q\neq 1 for 1/p+1/q=1+\alpha then the formula

\int\limits _{a}^{b}f(x)(\mathrm{I}_{{a+}}^{{\alpha}}g)(x)\mathrm{d}x=\int\limits _{a}^{b}g(x)(\mathrm{I}_{{b-}}^{{\alpha}}f)(x)\mathrm{d}x (2.60)

holds. [30.2.2] The formula is known as fractional integration by parts [99]. [30.2.3] For f(x)\in L^{{p}}(\mathbb{R}),g\in L^{{q}}(\mathbb{R}) with p>1,q>1 and 1/p+1/q=1+\alpha the analogous formula

\int\limits _{{-\infty}}^{{\infty}}f(x)(\mathrm{I}_{{+}}^{{\alpha}}g)(x)\mathrm{d}x=\int\limits _{{-\infty}}^{{\infty}}g(x)(\mathrm{I}_{{-}}^{{\alpha}}f)(x)\mathrm{d}x (2.61)

holds for Weyl fractional integrals.

[30.3.1] These formulae provide a second method of generalizing fractional integration to distributions. [30.3.2] Equation (2.60) may be read as

\langle\mathrm{I}_{{a+}}^{{\alpha}}f,\varphi\rangle=\langle f,\mathrm{I}_{{b-}}^{{\alpha}}\varphi\rangle (2.62)

for a distribution f and a test function \varphi. [30.3.3] It shows that right- and left-sided fractional integrals are adjoint operators. [30.3.4] The formula may be viewed as a definition of the fractional integral \mathrm{I}_{{a+}}^{{\alpha}}f of a distribution provided that the operator \mathrm{I}_{{b-}}^{{\alpha}} maps the test function space into itself.

2.2.1.8 Hardy-Littlewood Theorem

[30.4.1] The mapping properties of convolutions can be studied with the help of Youngs inequality. Let p,q,r obey 1\leq p,q,r\leq\infty and 1/p+1/q=1+1/r. [30.4.2] If K\in L^{{p}}(\mathbb{R}) and f\in L^{{q}}(\mathbb{R}) then K*f\in L^{{r}}(\mathbb{R}) and Youngs inequality \| K*f\| _{r}\leq\| K\| _{p}\;\| f\| _{q} holds. [30.4.3] It follows that \| K*f\| _{q}\leq C\| f\| _{p} if

[page 31, §0]    1\leq p\leq q\leq\infty and K\in L^{{r}}(\mathbb{R}) with 1/r=1+(1/q)-(1/p). [31.0.1] The Hardy-Littlewood theorem states that these estimates remain valid for K^{\alpha}_{\pm} although these kernels do not belong to any L^{{p}}(\mathbb{R})-space [37, 38]. [31.0.2] The theorem was generalized to higher dimensions by Sobolev in 1938, and is also known as the Hardy-Littlewood-Sobolev inequality (see [37, 38, 113, 63]).

Theorem 2.5

[31.1.1] Let 0<\alpha<1, 1<p<1/\alpha, -\infty\leq a<b\leq\infty. [31.1.2] Then \mathrm{I}_{{a+}}^{{\alpha}},\mathrm{I}_{{b-}}^{{\alpha}} are bounded linear operators from L^{{p}}([a,b]) to L^{{q}}([a,b]) with 1/q=(1/p)-\alpha,i.e. there exists a constant C(p,q) independent of f such that \|\mathrm{I}_{{a+}}^{{\alpha}}f\| _{q}\leq C\| f\| _{p}.

2.2.1.9 Additivity

[31.2.1] The basic composition law for fractional integrals follows from

\displaystyle(K_{+}^{\alpha}*K_{+}^{\beta})(x) \displaystyle=\int\limits _{0}^{x}K_{+}^{{\alpha}}(x-y)K_{+}^{\beta}(y)\;\mathrm{d}y=\int\limits _{0}^{{x}}\frac{(x-y)^{{\alpha-1}}}{\Gamma(\alpha)}\frac{y^{{\beta-1}}}{\Gamma(\beta)}\mathrm{d}y
\displaystyle=\frac{x^{{\alpha-1}}}{\Gamma(\alpha)}\frac{x^{{\beta-1}}}{\Gamma(\beta)}\int\limits _{0}^{1}(1-z)^{{\alpha-1}}z^{{\beta-1}}x\mathrm{d}z
\displaystyle=\frac{x^{{\alpha+\beta-1}}}{\Gamma(\alpha+\beta)}=K_{+}^{{\alpha+\beta}}(x), (2.63)

where Euler’s Beta-function

\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}=\int\limits _{0}^{1}(1-z)^{{\alpha-1}}z^{{\beta-1}}\mathrm{d}z=\mathrm{B}(\alpha,\beta) (2.64)

was used. [31.2.2] This implies the semigroup law for exponents

\mathrm{I}_{{a+}}^{{\alpha}}\mathrm{I}_{{a+}}^{{\beta}}=\mathrm{I}_{{a+}}^{{\alpha+\beta}}, (2.65)

also called additivity law. [31.2.3] It holds for Riemann-Liouville, Weyl and Riesz-Feller fractional integrals of functions.

2.2.2 Fractional Derivatives

2.2.2.1 Riemann-Liouville Fractional Derivatives

[31.3.1] Riemann [96, p.341] suggested to define fractional derivatives as integer order derivatives of fractional integrals.

Definition 2.6

[31.4.1] Let -\infty\leq a<x<b\leq\infty. [31.4.2] The Riemann-Liouville fractional derivative of order 0<\alpha<1 with lower limit a (resp. upper limit b) is defined for

[page 32, §0]    functions such that f\in L^{{1}}([a,b]) and f*K^{{1-\alpha}}\in W^{{1,1}}([a,b]) as

(\mathrm{D}^{{\alpha}}_{{a\pm}}f)(x)=\pm\frac{\mathrm{d}}{\mathrm{d}x}(\mathrm{I}_{{a\pm}}^{{1-\alpha}}f)(x) (2.66)

and (\mathrm{D}^{{0}}_{{a\pm}}f)(x)=f(x) for \alpha=0. [32.0.1] For \alpha>1 the definition is extended for functions f\in L^{{1}}([a,b]) with f*K^{{n-\alpha}}\in W^{{n,1}}([a,b]) as

(\mathrm{D}^{{\alpha}}_{{a\pm}}f)(x)=(\pm 1)^{n}\frac{\mathrm{d}^{{n}}}{\mathrm{d}x^{{n}}}(\mathrm{I}_{{a\pm}}^{{n-\alpha}}f)(x), (2.67)

where5 (This is a footnote:) 5[x] is the largest integer smaller than x. n=[\mathrm{Re}\,\alpha]+1 is smallest integer larger than \alpha.

[32.1.1] Here W^{{k,p}}(G)=\{ f\in L^{{p}}(G):\mathrm{D}^{k}f\in L^{{p}}(G)\} denotes a Sobolev space defined in (B.17). [32.1.2] For k=p=1 the space W^{{1,1}}([a,b])=\mathrm{AC}^{{0}}([a,b]) coincides with the space of absolutely continuous functions.

[32.2.1] The notation for fractional derivatives is not standardized6 (This is a footnote:) 6see footnote 2.1. [32.2.2] Leibniz and Euler used \mathrm{d}^{\alpha} [73, 72, 25] Riemann wrote \partial^{\alpha}_{x} [96], Liouville preferred \mathrm{d}^{\alpha}/\mathrm{d}x^{\alpha} [76], Grünwald used \{\mathrm{d}^{\alpha}f/\mathrm{d}x^{\alpha}\} _{{x=a}}^{{x=x}} or \mathrm{D}^{\alpha}[f]_{{x=a}}^{{x=x}} [34], Marchaud wrote \mathrm{D}^{{(\alpha)}}_{{a}}, and Hardy-Littlewood used an index f^{\alpha} [37]. [32.2.3] The notation in (2.67) follows [99, 98, 54, 52]. Modern authors also use I^{{-\alpha}} [97], I^{{-\alpha}}_{x} [23], {}_{a}D_{x}^{\alpha} [94, 85, 102], \mathrm{d}^{{\alpha}}/\mathrm{d}x^{{\alpha}} [129, 102], \mathrm{d}^{{\alpha}}/\mathrm{d}(x-a)^{{\alpha}} [92] instead of \mathrm{D}^{{\alpha}}_{{a+}}.

[32.3.1] Let f(x) be absolutely continuous on the finite interval [a,b]. [32.3.2] Then, its derivative f^{{\prime}} exists almost everywhere on [a,b] with f^{{\prime}}\in L^{{1}}([a,b]), and the function f can be written as

f(x)=\int\limits _{a}^{x}f^{{\prime}}(y)\mathrm{d}y+f(a)=(\mathrm{I}_{{a+}}^{{1}}f^{{\prime}})(x)+f(a). (2.68)

Substituting this into \mathrm{I}_{{a+}}^{{\alpha}}f gives

(\mathrm{I}_{{a+}}^{{\alpha}}f)(x)=(\mathrm{I}_{{a+}}^{{1}}\mathrm{I}_{{a+}}^{{\alpha}}f^{{\prime}})(x)+\frac{f(a)}{\Gamma(\alpha+1)}(x-a)^{{\alpha}}, (2.69)

where commutativity of \mathrm{I}_{{a+}}^{{1}} and \mathrm{I}_{{a+}}^{{\alpha}} was used. [32.3.3] It follows that

(\mathrm{D}\mathrm{I}_{{a+}}^{{\alpha}}f)(x)-(\mathrm{I}_{{a+}}^{{\alpha}}\mathrm{D}f)(x)=\frac{f(a)}{\Gamma(\alpha)}(x-a)^{{\alpha-1}} (2.70)

for 0<\alpha<1. [32.3.4] Above, the notations

(\mathrm{D}f)(x)=\frac{\mathrm{d}f(x)}{\mathrm{d}x}=f^{\prime}(x) (2.71)

were used for the first order derivative.

[page 33, §1]    [33.1.1] This observation suggests to introduce a modified Riemann-Liouville fractional derivative through

(\widetilde{\mathrm{D}}^{{\alpha}}_{{a+}}f)(x):=\mathrm{I}_{{a+}}^{{n-\alpha}}f^{{(n)}}(x)=\frac{1}{\Gamma(n-\alpha)}\int\limits _{a}^{x}\frac{f^{{(n)}}(y)}{(x-y)^{{\alpha-n+1}}}\mathrm{d}y, (2.72)

where n=[\mathrm{Re}\,\alpha]+1. [33.1.2] Note, that f must be at least n-times differentiable. [33.1.3] Formula (2.72) is due to Liouville [76, p.10] (see eq. (2.18) above), but nowadays sometimes named after Caputo [17].

[33.2.1] The relation between (2.72) and (2.67) is given by

Theorem 2.7

[33.2.2] For f\in\mathrm{AC}^{{n-1}}([a,b]) with n=[\mathrm{Re}\,\alpha]+1 the Riemann-Liouville fractional derivative (\mathrm{D}^{{\alpha}}_{{a+}}f)(x) exists almost everywhere for \mathrm{Re}\,\alpha\geq 0. [33.2.3] It can be written as

(\mathrm{D}^{{\alpha}}_{{a+}}f)(x)=(\widetilde{\mathrm{D}}^{{\alpha}}_{{a+}}f)(x)+\sum _{{k=0}}^{{n-1}}\frac{(x-a)^{{k-\alpha}}}{\Gamma(k-\alpha+1)}f^{{(k)}}(a) (2.73)

in terms of the Liouville(-Caputo) derivative defined in (2.72).

[33.3.1] The Riemann-Liouville fractional derivative is the left inverse of Riemann-Liouville fractional integrals. [33.3.2] More specifically, [99, p.44]

Theorem 2.8

[33.3.3] Let f\in L^{{1}}([a,b]). [33.3.4] Then

\mathrm{D}^{{\alpha}}_{{a+}}\mathrm{I}_{{a+}}^{{\alpha}}f(x)=f(x) (2.74)

holds for all \alpha with \mathrm{Re}\,\alpha\geq 0.

[33.4.1] For the right inverses of fractional integrals one finds

Theorem 2.9

[33.4.2] Let f\in L^{{1}}([a,b]) and \mathrm{Re}\,\alpha>0. [33.4.3] If in addition \mathrm{I}_{{a+}}^{{n-\alpha}}f\in\mathrm{AC}^{{n}}([a,b]) where n=[\mathrm{Re}\,\alpha]+1 then

\mathrm{I}_{{a+}}^{{\alpha}}\mathrm{D}^{{\alpha}}_{{a+}}f(x)=f(x)-\sum _{{k=0}}^{{n-1}}\frac{(x-a)^{{\alpha-k-1}}}{\Gamma(\alpha-k)}\left(\mathrm{D}^{{n-k-1}}\mathrm{I}_{{a+}}^{{n-\alpha}}f\right)(a) (2.75)

holds. [33.4.4] For 0<\mathrm{Re}\,\alpha<1 this becomes

\mathrm{I}_{{a+}}^{{\alpha}}\mathrm{D}^{{\alpha}}_{{a+}}f(x)=f(x)-\frac{(\mathrm{I}_{{a+}}^{{1-\alpha}}f)(a)}{\Gamma(\alpha)}(x-a)^{{\alpha-1}}. (2.76)

[33.5.1] The last theorem implies that for f\in L^{{1}}([a,b]) and \mathrm{Re}\,\alpha>0 with n=[\mathrm{Re}\,\alpha]+1 the equality

\mathrm{I}_{{a+}}^{{\alpha}}\mathrm{D}^{{\alpha}}_{{a+}}f(x)=f(x) (2.77)

[page 34, §0]    holds only if

\mathrm{I}_{{a+}}^{{n-\alpha}}f\in\mathrm{AC}^{{n}}([a,b])\\ (2.78a)
and
\left(\mathrm{D}^{k}\mathrm{I}_{{a+}}^{{n-\alpha}}f\right)(a)=0 (2.78b)

for all k=0,1,2,...,n-1. [34.0.1] Note that the existence of g(x)=\mathrm{D}^{{\alpha}}_{{a+}}f(x) in eq. (2.77) does not imply that f(x) can be written as (\mathrm{I}_{{a+}}^{{\alpha}}g)(x) for some integrable function g [99]. [34.0.2] This holds only if both conditions (2.78) are satisfied. [34.0.3] As an example where one of them fails, consider the function f(x)=(x-a)^{{\alpha-1}} for 0<\alpha<1. [34.0.4] Then \mathrm{D}^{{\alpha}}_{{a+}}(x-a)^{{\alpha-1}}=0 exists. [34.0.5] Now \mathrm{D}^{0}\mathrm{I}_{{a+}}^{{1-\alpha}}(x-a)^{{\alpha-1}}\neq 0 so that (2.78b) fails. [34.0.6] There does not exist an integrable g such that \mathrm{I}_{{a+}}^{{\alpha}}g=(x-a)^{{\alpha-1}}. [34.0.7] In fact, g corresponds to the \delta-distribution \delta(x-a).

2.2.2.2 General Types of Fractional Derivatives

[34.1.1] Riemann-Liouville fractional derivatives have been generalized in [52, p.433] to fractional derivatives of different types.

Definition 2.10

[34.2.1] The generalized Riemann-Liouville fractional derivative of order 0<\alpha<1 and type 0\leq\beta\leq 1 with lower (resp. upper) limit a is defined as

(\mathrm{D}^{{\alpha,\beta}}_{{a\pm}}f)(x)=\left(\pm\mathrm{I}_{{a\pm}}^{{\beta(1-\alpha)}}\frac{\mathrm{d}}{\mathrm{d}x}\left(\mathrm{I}_{{a\pm}}^{{(1-\beta)(1-\alpha)}}f\right)\right)(x) (2.79)

for functions such that the expression on the right hand side exists.

[34.3.1] The type \beta of a fractional derivative allows to interpolate continuously from \mathrm{D}^{{\alpha}}_{{a\pm}}=\mathrm{D}^{{\alpha,0}}_{{a\pm}} to \widetilde{\mathrm{D}}^{{\alpha}}_{{a\pm}}=\mathrm{D}^{{\alpha,1}}_{{a\pm}}. [34.3.2] A relation between fractional derivatives of the same order but different types was given in [52, p.434].

2.2.2.3 Marchaud-Hadamard Fractional Derivatives

[34.4.1] Marchaud’s approach [78] is based on Hadamards finite parts of divergent integrals [36]. [34.4.2] The strategy is to define fractional derivatives as analytic continuation of fractional integrals to negative orders. [see [99, p.225]]

Definition 2.11

[34.5.1] Let -\infty<a<b<\infty and 0<\alpha<1. [34.5.2] The Marchaud fractional derivative of order \alpha with lower limit a is defined as

(\mathrm{M}^{{\alpha}}_{{a+}}f)(x)=\frac{f(x)}{\Gamma(1-\alpha)(x-a)^{\alpha}}+\frac{\alpha}{\Gamma(1-\alpha)}\int\limits _{a}^{x}\frac{f(x)-f(y)}{(x-y)^{{\alpha+1}}}\mathrm{d}y (2.80)

[page 35, §0]    and the Marchaud fractional derivative of order \alpha with upper limit b is defined as

(\mathrm{M}^{{\alpha}}_{{b-}}f)(x)=\frac{f(x)}{\Gamma(1-\alpha)(b-x)^{\alpha}}+\frac{\alpha}{\Gamma(1-\alpha)}\int\limits _{x}^{b}\frac{f(x)-f(y)}{(x-y)^{{\alpha+1}}}\mathrm{d}y. (2.81)

[35.0.1] For a=-\infty (resp. b=\infty) the definition is

(\mathrm{M}^{{\alpha}}_{{\pm}}f)(x)=\frac{\alpha}{\Gamma(1-\alpha)}\int\limits _{0}^{\infty}\frac{f(x)-f(x\mp y)}{y^{{\alpha+1}}}\mathrm{d}y. (2.82)

[35.0.2] The definition is completed with \mathrm{M}^{{0}}f=f for all variants.

[35.1.1] The idea of Marchaud’s method is to extend the Riemann-Liouville integral from \alpha>0 to \alpha<0, and to define

(\mathrm{I}_{{+}}^{{-\alpha}}f)(x)=\frac{1}{\Gamma(-\alpha)}\int\limits _{0}^{\infty}y^{{-\alpha-1}}f(x-y)\;\mathrm{d}y, (2.83)

where \alpha>0. [35.1.2] However, this is not possible because the integral in (2.83) diverges. [35.1.3] The idea is to subtract the divergent part of the integral,

\int _{\varepsilon}^{\infty}y^{{-\alpha-1}}f(x)\mathrm{d}y=\frac{f(x)}{\alpha\varepsilon^{\alpha}} (2.84)

obtained by setting f(x-y)\approx f(x) for y\approx 0. [35.1.4] Subtracting (2.83) from (2.84) for 0<\alpha<1 suggests the definition

(\mathrm{M}^{{\alpha}}_{{+}}f)(x)=\lim _{{\varepsilon\to 0+}}\frac{1}{\Gamma(-\alpha)}\int\limits _{\varepsilon}^{\infty}\frac{f(x)-f(x-y)}{y^{{\alpha+1}}}\;\mathrm{d}y (2.85)

[35.1.5] Formal integration by parts leads to (\mathrm{I}_{{+}}^{{1-\alpha}}f^{{\prime}})(x), showing that this definition contains the Riemann-Liouville definition.

[35.2.1] The definition may be extended to \alpha>1 in two ways. [35.2.2] The first consists in applying (2.85) to the n-th derivative \mathrm{d}^{n}f/\mathrm{d}x^{n} for n<\alpha<n+1. [35.2.3] The second possibility is to regard f(x-y)-f(x) as a first order difference, and to generalize to n-th order differences. [35.2.4] The n-th order difference is

(\Delta^{n}_{y}f)(x)=(\mathbf{1}-\mathsf{T}_{y})^{n}f(x)=\sum _{{k=0}}^{n}(-1)^{k}\binom{n}{k}f(x-ky), (2.86)

where (\mathbf{1}f)(x)=f(x) is the identity operator and

(\mathsf{T}_{h}f)(x)=f(x-h) (2.87)

[page 36, §0]    is the translation operator. [36.0.1] The Marchaud fractional derivative can then be extended to 0<\alpha<n through [94, 98]

(\mathrm{M}^{{\alpha}}_{{+}}f)(x)=\lim _{{\varepsilon\to 0+}}\frac{1}{C_{{\alpha,n}}}\int\limits _{\varepsilon}^{\infty}\frac{\Delta^{n}_{y}f(x)}{y^{{\alpha+1}}}\;\mathrm{d}y, (2.88)

where

C_{{\alpha,n}}=\int\limits _{0}^{\infty}\frac{(1-\mathrm{e}^{{-y}})^{n}}{y^{{\alpha+1}}}\mathrm{d}y, (2.89)

where the limit may be taken in the sense of pointwise or norm convergence.

[36.1.1] The Marchaud derivatives \mathrm{M}^{{\alpha}}_{{\pm}} are defined for a wider class of functions than Weyl derivatives \mathrm{D}^{{\alpha}}_{{\pm}}. [36.1.2] As an example consider the function f(x)=const.

[36.2.1] Let f be such that there exists a function g\in L^{{1}}([a,b]) with f=\mathrm{I}_{{a+}}^{{\alpha}}g. [36.2.2] Then the Riemann-Liouville derivative and the Marchaud derivative coincide almost everywhere, i.e. (\mathrm{M}^{{\alpha}}_{{a+}}f)(x)=(\mathrm{D}^{{\alpha}}_{{a+}}f)(x) for almost all x [99, p.228].

2.2.2.4 Weyl Fractional Derivatives

[36.3.1] There are two kinds of Weyl fractional derivatives for periodic functions. [36.3.2] The Weyl-Liouville fractional derivative is defined as [99, p.351],[94]

(\mathrm{D}^{{\alpha}}_{{\pm}}f)(x)=\pm\frac{\mathrm{d}}{\mathrm{d}x}(\mathrm{I}_{{\pm}}^{{1-\alpha}}f)(x) (2.90)

for 0<\alpha<1 where the Weyl integral \pm\mathrm{I}_{{\pm}}^{{\alpha}}f was defined in (2.34). [36.3.3] The Weyl-Marchaud fractional derivative is defined as [99, p.352],[94]

(\mathrm{W}^{{\alpha}}_{{\pm}}f)(x)=\frac{1}{2\pi}\int\limits _{0}^{{2\pi}}\left[f(x-y)-f(x)\right](\mathrm{D}^{1}\Psi _{\pm}^{{1-\alpha}})(y)\mathrm{d}y (2.91)

for 0<\alpha<1 where \Psi _{\pm}(x) is defined in eq. (2.35). [36.3.4] The Weyl derivatives are defined for periodic functions of with zero mean in C^{{\beta}}(\mathbb{R}/2\pi\mathbb{Z}) where \beta>\alpha. [36.3.5] In this space (\mathrm{D}^{{\alpha}}_{{\pm}}f)(x)=(\mathrm{W}^{{\alpha}}_{{\pm}}f)(x), i.e. the Weyl-Liouville and Weyl-Marchaud form coincide [99]. [36.3.6] As for fractional integrals, it can be shown that the Weyl-Liouville derivative (0<\alpha<1)

(\mathrm{D}^{{\alpha}}_{{+}}f)(x)=\frac{1}{\Gamma(1-\alpha)}\int\limits _{{-\infty}}^{x}\frac{f(y)}{(x-y)^{\alpha}}\mathrm{d}y (2.92)

coincides with the Riemann-Liouville derivative with lower limit -\infty. [36.3.7] In addition one has the equivalence \mathrm{D}^{{\alpha}}_{{+}}f=\mathrm{W}^{{\alpha}}_{{+}}f with the Marchaud-Hadamard fractional derivative in a suitable sense [99, p.357].

[page 37, §1]

2.2.2.5 Riesz Fractional Derivatives

[37.1.1] To define the Riesz fractional derivative as integer derivatives of Riesz potentials consider the Fourier transforms

{\mathcal{F}}\left\{\mathrm{D}\mathrm{I}^{{1-\alpha}}f\right\}(k)=(\mathrm{i}k)|k|^{{\alpha-1}}{\mathcal{F}}\left\{ f\right\}(k)=(\mathrm{i}\mathrm{sgn}\, k)|k|^{\alpha}{\mathcal{F}}\left\{ f\right\}(k) (2.93)
{\mathcal{F}}\left\{\mathrm{D}\widetilde{\mathrm{I}^{{1-\alpha}}}f\right\}(k)=(\mathrm{i}k)(-\mathrm{i}\mathrm{sgn}\, k)|k|^{{\alpha-1}}{\mathcal{F}}\left\{ f\right\}(k)=|k|^{\alpha}{\mathcal{F}}\left\{ f\right\}(k) (2.94)

for 0<\alpha<1. [37.1.2] Comparing this to eq. (2.57) suggests to consider

\frac{\mathrm{d}}{\mathrm{d}x}(\widetilde{\mathrm{I}^{{1-\alpha}}}f)(x)=\lim _{{h\to 0}}\frac{1}{h}\left[(\widetilde{\mathrm{I}^{{1-\alpha}}}f)(x+h)-(\widetilde{\mathrm{I}^{{1-\alpha}}}f)(x)\right] (2.95)

as a candidate for the Riesz fractional derivative.

[37.2.1] Following [94] the strong Riesz fractional derivative of order \alpha \mathrm{R}^{{\alpha}}f of a function f\in L^{{p}}(\mathbb{R}), 1\leq p<\infty, is defined through the limit

\lim _{{h\to 0}}\left\|\frac{1}{h}(f*K^{{1-\alpha}}_{h})-\mathrm{R}^{{\alpha}}f\right\| _{p}=0, (2.96)

whenever it exists. [37.2.2] The convolution kernel defined as

K_{h}^{{1-\alpha}}=\frac{1}{2\Gamma(1-\alpha)\sin(\alpha\pi/2)}\left[\frac{\mathrm{sgn}\,(x+h)}{|x+h|^{\alpha}}-\frac{\mathrm{sgn}\, x}{|x|^{\alpha}}\right] (2.97)

is obtained from eq. (2.95). [37.2.3] Indeed, this definition is equivalent to eq. (2.94). [37.2.4] A function f\in L^{{p}}(\mathbb{R}) where 1\leq p\leq 2 has a strong Riesz derivative of order \alpha if and only if there exsists a function g\in L^{{p}}(\mathbb{R}) such that |k|^{\alpha}{\mathcal{F}}\left\{ f\right\}(k)={\mathcal{F}}\left\{ g\right\}(k). [37.2.5] Then \mathrm{R}^{{\alpha}}f=g.

2.2.2.6 Grünwald-Letnikov Fractional Derivatives

[37.3.1] The basic idea of the Grünwald approach is to generalize finite difference quotients to noninteger order, and then take the limit to obtain a differential quotient. [37.3.2] The first order derivative is the limit

\frac{\mathrm{d}}{\mathrm{d}x}f(x)=(\mathrm{D}f)(x)=\lim _{{h\to 0}}\frac{f(x)-f(x-h)}{h}=\lim _{{h\to 0}}\frac{[\mathbf{1}-\mathsf{T}(h)]}{h}f(x) (2.98)

of a difference quotient. [37.3.3] In the last equality (\mathbf{1}f)(x)=f(x) is the identity operator, and

[\mathsf{T}(h)f](x)=f(x-h) (2.99)

is the translation operator. [37.3.4] Repeated application of \mathsf{T} gives

[\mathsf{T}(h)^{n}f](x)=f(x-nh), (2.100)

[page 38, §0]    where n\in\mathbb{N}. [38.0.1] The second order derivative can then be written as

\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}f(x)=(\mathrm{D}^{2}f)(x) \displaystyle=\lim _{{h\to 0}}\frac{f(x)-2f(x-h)+f(x-2h)}{h^{2}}
\displaystyle=\lim _{{h\to 0}}\left\{\frac{[\mathbf{1}-\mathsf{T}(h)]}{h}\right\}^{2}f(x), (2.101)

and the n-th derivative

\displaystyle\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}f(x)=(\mathrm{D}^{n}f)(x) \displaystyle=\lim _{{h\to 0}}\frac{1}{h^{n}}\sum _{{k=0}}^{n}(-1)^{k}\binom{n}{k}f(x-kh)
\displaystyle=\lim _{{h\to 0}}\left\{\frac{[\mathbf{1}-\mathsf{T}(h)]}{h}\right\}^{n}f(x), (2.102)

which exhibits the similarity with the binomial formula. [38.0.2] The generalization to noninteger n gives rise to fractional difference quotients defined through

(\Delta^{\alpha}_{h}f)(x)=\sum _{{k=0}}^{\infty}(-1)^{k}\binom{\alpha}{k}f(x-kh) (2.103)

for \alpha>0. [38.0.3] These are generally divergent for \alpha<0. [38.0.4] For example, if f(x)=1, then

\sum _{{k=0}}^{N}(-1)^{k}\binom{\alpha}{k}=\frac{1}{\Gamma(1-\alpha)}\frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)} (2.104)

diverges as N\to\infty if \alpha<0. [38.0.5] Fractional difference quotients were studied in [68]. Note that fractional differences obey [99]

(\Delta^{\alpha}_{h}(\Delta^{\beta}_{h}f))(x)=(\Delta^{{\alpha+\beta}}_{h}f)(x). (2.105)
Definition 2.12

[38.1.1] The Grünwald-Letnikov fractional derivative of order \alpha>0 is defined as the limit

(\mathrm{G}^{{\alpha}}_{{\pm}}f)(x)=\lim _{{h\to 0^{+}}}\frac{1}{h^{\alpha}}(\Delta^{\alpha}_{{\pm h}}f)(x) (2.106)

of fractional difference quotients whenever the limit exists. [38.1.2] The Grünwald Letnikov fractional derivative is called pointwise or strong depending on whether the limit is taken pointwise or in the norm of a suitable Banach space.

[38.2.1] For a definition of Banach spaces and their norms see e.g. [128].

[38.3.1] The Grünwald-Letnikov fractional derivative has been studied for periodic functions in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}) with 1\leq p<\infty in [99, 94]. [38.3.2] It has the following properties.

[page 39, §1]   

Theorem 2.13

[39.1.1] Let f\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}), 1\leq p<\infty and \alpha>0. [39.1.2] Then the following statements are equivalent:

  1. \mathrm{G}^{{\alpha}}_{{+}}f\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z})

  2. [39.1.3] There exists a function g\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}) such that
    (\mathrm{i}k)^{\alpha}{\mathcal{F}}\left\{ f(x)\right\}(k)={\mathcal{F}}\left\{ g(x)\right\}(k) where k\in\mathbb{Z}.

  3. [39.1.4] There exists a function g\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}) such that
    f(x)-{\mathcal{F}}\left\{ f(x)\right\}(0)=(\mathrm{I}_{{+}}^{{\alpha}}g)(x) holds for almost all x.

Theorem 2.14

[39.2.1] Let f\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}), 1\leq p<\infty and \alpha,\beta>0. [39.2.2] Then:

  1. \mathrm{G}^{{\alpha}}_{{+}}f\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}) implies \mathrm{G}^{{\beta}}_{{+}}f\in L^{{p}}(\mathbb{R}/2\pi\mathbb{Z}) for every 0<\beta<\alpha.

  2. \mathrm{G}^{{\alpha}}_{{+}}\mathrm{G}^{{\beta}}_{{+}}f=\mathrm{G}^{{\alpha+\beta}}_{{+}}f

  3. \mathrm{G}^{{\alpha}}_{{+}}(\mathrm{I}_{{+}}^{{\alpha}}f)=f(x)-{\mathcal{F}}\left\{ f\right\}(0)

2.2.2.7 Fractional Derivatives of Distributions

[39.3.1] The basic idea for defining fractional differentiation of distributions is to extend the definition of fractional integration (2.54) to negative \alpha. [39.3.2] However, for \mathrm{Re}\,\alpha<0 the distribution K_{+}^{\alpha} becomes singular because x^{{\alpha-1}} is not locally integrable in this case. [39.3.3] The extension of K_{+}^{\alpha} to \mathrm{Re}\,\alpha<0 requires regularization [31, 128, 63]. [39.3.4] It turns out that the regularization exists and is essentially unique as long as (-\alpha)\notin\mathbb{N}_{0}.

Definition 2.15

[39.4.1] Let f be a distribution f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, f\subset\mathbb{R}_{+}. [39.4.2] Then the fractional derivative of order \alpha with lower limit 0 is the distribution \mathrm{D}^{{\alpha}}_{{0+}}f defined as

\langle\mathrm{D}^{{\alpha}}_{{0+}}f,\varphi\rangle=\langle\mathrm{D}^{{\alpha}}_{{+}}f,\varphi\rangle=\langle K_{+}^{{-\alpha}}*f,\varphi\rangle, (2.107)

where \alpha\in\mathbb{C} and

K_{+}^{\alpha}(x)=\begin{cases}\Theta(x)\displaystyle\frac{x^{{\alpha-1}}}{\Gamma(\alpha)}&,\mathrm{Re}\,\alpha>0\\ &\\ \displaystyle\frac{\mathrm{d}^{N}}{\mathrm{d}x^{N}}\left[\Theta(x)\displaystyle\frac{x^{{\alpha+N-1}}}{\Gamma(\alpha+N)}\right]&,\mathrm{Re}\,\alpha+N>0,N\in\mathbb{N}\end{cases} (2.108)

is the kernel distribution. [39.4.3] For \alpha=0 one finds K_{+}^{0}(x)=(\mathrm{d}/\mathrm{d}x)\Theta(x)=\delta(x) and \mathrm{D}^{{0}}_{{0+}}=\mathbf{1} as the identity operator. [39.4.4] For the \alpha=-k,k\in\mathbb{N} one finds

K_{+}^{{-k}}(x)=\delta^{{(k)}}(x), (2.109)

where \delta^{{(k)}} is the k-th derivative of the \delta distribution.

[page 40, §1]    [40.1.1] The kernel distribution in (2.108) is

K_{+}^{{-\alpha}}(x)=\frac{\mathrm{d}}{\mathrm{d}x}\left[\Theta(x)\frac{x^{{-\alpha}}}{\Gamma(1-\alpha)}\right]=\frac{\mathrm{d}}{\mathrm{d}x}K_{+}^{{1-\alpha}}(x) (2.110)

for 0<\alpha<1. [40.1.2] Its regularized action is

\displaystyle\left\langle K_{+}^{{-\alpha}}(x),\varphi(x)\right\rangle \displaystyle=\left\langle\frac{\mathrm{d}}{\mathrm{d}x}K_{+}^{{1-\alpha}}(x),\varphi(x)\right\rangle=-\left\langle K_{+}^{{1-\alpha}}(x),\varphi(x)^{\prime}\right\rangle (2.111a)
\displaystyle=-\frac{1}{\Gamma(1-\alpha)}\lim _{{\varepsilon\to 0}}\int\limits _{\varepsilon}^{\infty}x^{{-\alpha}}\varphi(x)^{\prime}\mathrm{d}x (2.111b)
\displaystyle=-\lim _{{\varepsilon\to 0}}\left\{\left.\frac{\varphi(x)+C}{\Gamma(\alpha)x^{{\alpha}}}\right|_{{\varepsilon}}^{\infty}-\int\limits _{{\varepsilon}}^{\infty}\frac{\varphi(x)+C}{\Gamma(-\alpha)x^{{1+\alpha}}}\mathrm{d}x\right\} (2.111c)
\displaystyle=\int\limits _{0}^{\infty}\frac{\varphi(x)-\varphi(0)}{\Gamma(-\alpha)x^{{1+\alpha}}}\mathrm{d}x, (2.111d)

where \varphi(\infty)<\infty was assumed in the last step and the arbitrary constant was chosen as C=-\varphi(0). [40.1.3] This choice regularizes the divergent first term in (2.111c). [40.1.4] If this rule is used for the distributional convolution

(K_{+}^{{-\alpha}}*f)(x)=\frac{1}{\Gamma(-\alpha)}\int\limits _{0}^{\infty}\frac{f(x)-f(x-y)}{y^{{\alpha+1}}}\mathrm{d}y=(\mathrm{M}^{{\alpha}}_{{+}}f)(x) (2.112)

then the Marchaud-Hadamard form is recovered with 0<\alpha<1.

[40.2.1] It is now possible to show that the convolution of distributions is in general not associative. [40.2.2] A counterexample is

(1*\delta^{\prime})*\Theta=1^{\prime}*\Theta=0*\Theta=0\neq 1=1*\delta=1*\Theta^{\prime}=1*(\delta^{\prime}*\Theta), (2.113)

where \Theta is the Heaviside step function.

[40.3.1] \mathrm{D}^{{\alpha}}_{{0+}}f has support in \mathbb{R}_{+}. [40.3.2] The distributions in f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, f\subset\mathbb{R}_{+} form a convolution algebra [21] and one finds [31, 99]

Theorem 2.16

[40.3.3] If f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, f\subset\mathbb{R}_{+} then also \mathrm{I}_{{0+}}^{{\alpha}}f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{I}_{{0+}}^{{\alpha}}\mathrm{supp}\, f\subset\mathbb{R}_{+}. [40.3.4] Moreover, for all \alpha,\beta\in\mathbb{C}

\mathrm{D}^{{\alpha}}_{{0+}}\mathrm{D}^{{\beta}}_{{0+}}f=\mathrm{D}^{{\alpha+\beta}}_{{0+}}f (2.114)

with \mathrm{D}^{{\alpha}}_{{0+}}f=\mathrm{I}_{{0+}}^{{-\alpha}}f for \mathrm{Re}\,\alpha<0. [40.3.5] For each f\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, f\subset\mathbb{R}_{+} there exists a unique distribution g\in C_{{\mathrm{0}}}^{{\infty}}(\mathbb{R})^{\prime} with \mathrm{supp}\, g\subset\mathbb{R}_{+} such that f=\mathrm{I}_{{0+}}^{{\alpha}}g.

[page 41, §1]    [41.1.1] Note that

\mathrm{D}^{{\alpha}}_{{0+}}f=\mathrm{D}^{{\alpha}}_{{0+}}(\mathbf{1}f)=(K_{+}^{{-\alpha}}*K_{+}^{0})*f=(\mathrm{D}^{{\alpha}}_{{0+}}\delta)*f=\delta^{{(\alpha)}}*f (2.115)

for all \alpha\in\mathbb{C}. [41.2.1] Also, the differentiation rule

\mathrm{D}^{{\alpha}}_{{0+}}K_{+}^{\beta}=K_{+}^{{\beta-\alpha}} (2.116)

holds for all \alpha,\beta\in\mathbb{C}. [41.2.2] It contains

\mathrm{D}K_{+}^{\beta}=K_{+}^{{\beta-1}} (2.117)

for all \beta\in\mathbb{C} as a special case.

2.2.2.8 Fractional Derivatives at Their Lower Limit

[41.3.1] All fractional derivatives defined above are nonlocal operators. [41.3.2] A local fractional derivative operator was introduced in [40, 41, 52].

Definition 2.17

[41.4.1] For -\infty<a<\infty the Riemann-Liouville fractional derivative of order 0<\alpha<1 at the lower limit a is defined by

\left.\frac{\mathrm{d}^{\alpha}f}{\mathrm{d}x^{\alpha}}\right|_{{x=a}}=f^{{(\alpha)}}(a)=\lim _{{x\to a\pm}}(\mathrm{D}^{{\alpha}}_{{a\pm}}f)(x), (2.118)

whenever the two limits exist and are equal. [41.4.2] If f^{{(\alpha)}}(a) exists the function f is called fractionally differentiable at the limit a.

[41.5.1] These operators are useful for the analysis of singularities. [41.5.2] They were applied in [40, 41, 42, 44, 52] to the analysis of singularities in the theory of critical phenomena and to the generalization of Ehrenfests classification of phase transitions. [41.5.3] There is a close relationship to the theory of regularly varying functions [107] as evidenced by the following result [52].

Theorem 2.18

[41.6.1]   Let the function f:[0,\infty[\to\mathbb{R} be monotonously increasing with f(x)\geq 0 and f(0)=0, and such that (\mathrm{D}^{{\alpha,\lambda}}_{{0+}}f)(x) with 0<\alpha<1 and 0\leq\lambda\leq 1 is also monotonously increasing on a neighbourhood [0,\delta] for small \delta>0. [41.6.2] Let 0\leq\beta<\lambda(1-\alpha)+\alpha, let C\geq 0 be a constant and \Lambda(x) a slowly varying function for x\to 0. [41.6.3] Then

\lim _{{x\to 0}}\frac{f(x)}{x^{\beta}\Lambda(x)}=C (2.119)

holds if and only if

\lim _{{x\to 0}}\frac{(\mathrm{D}^{{\alpha,\lambda}}_{{0+}}f)(x)}{x^{{\beta-\alpha}}\Lambda(x)}=C\frac{\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)} (2.120)

holds.

[page 42, §1]

[42.1.1] A function f is called slowly varying at infinity if \lim _{{x\to\infty}}f(bx)/f(x)=1 for all b>0. [42.1.2] A function f(x) is called slowly varying at a\in\mathbb{R} if f(1/(x-a)) is slowly varying at infinity.

2.2.2.9 Fractional Powers of Operators

[42.2.1] The spectral decomposition of selfadjoint operators is a familiar mathematical tool from quantum mechanics [116]. [42.2.2] Let A denote a selfadjoint operator with domain D(A) and spectral family E_{\lambda} on a Hilbert space X with scalar product (\cdot,\cdot). [42.2.3] Then

(Au,v)=\int\limits _{{\sigma(A)}}\lambda\mathrm{d}(E_{\lambda}u,v) (2.121)

holds for all u,v\in D(A). [42.2.4] Here \sigma(A) is the spectrum of A. [42.2.5] It is then straightforward to define the fractional power A^{\alpha}u by

(A^{\alpha}u,u)=\int\limits _{{\sigma(A)}}\lambda^{\alpha}\mathrm{d}(E_{\lambda}u,u) (2.122)

on the domain

D(A^{\alpha})=\{ u\in X:\int\limits _{{\sigma(A)}}\lambda^{\alpha}\mathrm{d}(E_{\lambda}u,u)<\infty\}. (2.123)

[42.2.6] Similarly, for any measurable function g:\sigma(A)\to\mathbb{C} the operator g(A) is defined with an integrand g(\lambda) in eq. (2.122). [42.2.7] This yields an operator calculus that allows to perform calculations with functions instead of operators.

[42.3.1] Fractional powers of the Laplacian as the generator of the diffusion semigroup were introduced by Bochner [13] and Feller [26] based on Riesz’ fractional potentials. [42.3.2] The fractional diffusion equation

\frac{\partial f}{\partial t}=-(-\Delta)^{{\alpha/2}}f (2.124)

was related by Feller to the Levy stable laws [74] using one dimensional fractional integrals \mathrm{I}^{{-\alpha,\beta}} of order -\alpha and type \beta [26]7 (This is a footnote:) 7Fellers motivation to introduce the type \beta was this relation.. [42.3.3] For \alpha=2 eq. (2.124) reduces to the diffusion equation. [42.3.4] This type of fractional diffusion will be referred to as fractional diffusion of Bochner-Levy type (see Section 2.3.4 for more discussion). [42.3.5] Later, these ideas were extended to fractional powers of closed8 (This is a footnote:) 8 An operator A:B\to B on a Banach space B is called closed if the set of pairs (x,Ax) with x\in D(A) is closed in B\times B. semigroup generators [4, 5, 69, 70]. [42.3.6] If (-\mathrm{A}) is the infinitesimal generator of a

[page 43, §0]    semigroup T(t) (see Section 2.3.3.2 for definitions of T(t) and \mathrm{A}) on a Banach space B then its fractional power is defined as

(-\mathrm{A})^{\alpha}f=\lim _{{\varepsilon\to 0+}}\frac{1}{-\Gamma(-\alpha)}\int\limits _{\varepsilon}^{\infty}t^{{-\alpha-1}}[\mathbf{1}-T(t)]f\mathrm{d}t (2.125)

for every f\in B for which the limit exists in the norm of B [120, 121, 93, 123]. [43.0.1] This aproach is clearly inspired by the Marchaud form (2.82). Alternatively, one may use the Grünwald approach to define fractional powers of semigroup generators [122, 99].

2.2.2.10 Pseudodifferential Operators

[43.1.1] The calculus of pseudodifferential operators represents another generalization of the operator calculus in Hilbert spaces. [43.1.2] It has its roots in Hadamard’s ideas [36], Riesz potentials [97], Feller’s suggestion [26] and Calderon-Zygmund singular integrals [16]. [43.1.3] Later it was generalized and became a tool for treating elliptic partial differential operators with nonconstant coefficients.

Definition 2.19

[43.2.1] A (Kohn-Nirenberg) pseudodifferential operator of order \alpha\in\mathbb{R} \sigma(x,\mathrm{D}):\mathcal{S}{(\mathbb{R}^{d})}\to\mathcal{S}{(\mathbb{R}^{d})} is defined as

\sigma(x,\mathrm{D})f(x)=\frac{1}{(2\pi)^{d}}\int\limits _{{\mathbb{R}^{d}}}\mathrm{e}^{{\mathrm{i}xk}}\sigma(x,k){\mathcal{F}}\left\{ f\right\}(k)\mathrm{d}k (2.126)

and the function \sigma(x,k) is called its symbol. [43.2.2] The symbol is in the Kohn-Nirenberg symbol class S^{\alpha} if it is in C^{{\infty}}(\mathbb{R}^{{2d}}), and there exists a compact set K\subset\mathbb{R}^{d} such that \mathrm{supp}\,\sigma\subset K\times\mathbb{R}^{d}, and for any pair of multiindices \beta,\gamma there is a constant C_{{\beta,\gamma}} such that

\mathrm{D}_{k}^{\beta}\mathrm{D}_{x}^{\gamma}\sigma(x,k)\leq C_{{\beta,\gamma}}(1+|k|)^{{\alpha-|\beta|}}. (2.127)

[43.2.3] The Hörmander symbol class S^{\alpha}_{{\rho,\delta}} is obtained by replacing the exponent \alpha-|\beta| on the right hand side with \alpha-\rho|\beta|+\delta|\gamma| where 0\leq\rho,\delta\leq 1.

[43.3.1] Pseudodifferential operators provide a unified approach to differential and integral or convolution operators that are ‘‘nearly’’ translation invariant. [43.3.2] They have a close relation with Weyl quantization in physics [116, 28]. However, they will not be discussed further because the traditional symbol classes do not contain the usual fractional derivative operators. [43.3.3] Fractional Riesz derivatives are not pseudodifferential operators in the sense above. [43.3.4] Their symbols do not fall into any of the standard Kohn-Nirenberg or Hörmander symbol classes due to lack of differentiability at the origin.

[page 44, §1]

2.2.3 Eigenfunctions

[44.1.1] The eigenfunctions of Riemann-Liouville fractional derivatives are defined as the solutions of the fractional differential equation

(\mathrm{D}^{{\alpha}}_{{0+}}f)(x)=\lambda f(x), (2.128)

where \lambda is the eigenvalue. [44.1.2] They are readily identifed using eq. (A.11) as

f(x)=x^{{1-\alpha}}\mathrm{E}_{{\alpha,\alpha}}(\lambda x^{\alpha}), (2.129)

where

\mathrm{E}_{{\alpha,\beta}}=\sum _{{k=0}}^{\infty}\frac{x^{k}}{\Gamma(\alpha k+\beta)} (2.130)

is the generalized Mittag-Leffler function [125, 126]. [44.1.3] More generally the eigenvalue equation for fractional derivatives of order \alpha and type \beta reads

(\mathrm{D}^{{\alpha,\beta}}_{{0+}}f)(x)=\lambda f(x), (2.131)

and it is solved by [54, eq.124]

f(x)=x^{{(1-\beta)(1-\alpha)}}\mathrm{E}_{{\alpha,\alpha+\beta(1-\alpha)}}(\lambda x^{\alpha}), (2.132)
Figure 2.1: Truncated real part of the generalized Mittag-Leffler function -3\leq\mathrm{Re}\,\mathrm{E}_{{0.8,0.9}}(z)\leq 3 for z\in\mathbb{C} with -7\leq\mathrm{Re}\, z\leq 5 and -10\leq\mathrm{Im}\, z\leq 10. The solid line is defined by \mathrm{Re}\,\mathrm{E}_{{0.8,0.9}}(z)=0.

[page 45, §0]    where the case \beta=0 corresponds to (2.128). [45.0.1] A second important special case is the equation

(\mathrm{D}^{{\alpha,1}}_{{0+}}f)(x)=\lambda f(x), (2.133)

with \mathrm{D}^{{\alpha,1}}_{{0+}}=\widetilde{\mathrm{D}}^{{\alpha}}_{{0+}}. [45.0.2] In this case the eigenfunction

f(x)=\mathrm{E}_{\alpha}(\lambda x^{\alpha}), (2.134)

where \mathrm{E}_{\alpha}(x)=\mathrm{E}_{{\alpha,1}}(x) is the Mittag-Leffler function [86]. [45.0.3] The Mittag-Leffler function plays a central role in fractional calculus. [45.0.4] It has only recently been calculated numerically in the full complex plane [108, 62]. [45.0.5] Figure 2.1 and 2.2 illustrate \mathrm{E}_{{0.8,0.9}}(z) for a rectangular region in the complex plane (see [108]). [45.1.1] The solid line in Figure 2.1 is the line \mathrm{Re}\,\mathrm{E}_{{0.8,0.9}}(z)=0, in Figure 2.2 it is \mathrm{Im}\,\mathrm{E}_{{0.8,0.9}}(z)=0.

Figure 2.2: Same as Fig. 2.1 for the imaginary part of \mathrm{E}_{{0.8,0.9}}(z). The solid line is \mathrm{Im}\,\mathrm{E}_{{0.8,0.9}}(z)=0.

[45.2.1] Note, that some authors are avoiding the operator \mathrm{D}^{{\alpha,1}}_{{0+}} in fractional differential equations (see e.g. [112, 101, 84, 111, 7, 82] or chapters in this volume). [45.2.2] In their notation the eigenvalue equation (2.133) becomes (c.f.[112, eq.(22)])

\frac{\mathrm{d}}{\mathrm{d}x}f(x)=\lambda\mathrm{D}^{{1-\alpha}}_{{0+}}f(x) (2.135)

containing two derivative operators instead of one.