Mainardi, Francesco - Fractional Calculus and Waves in Liner Viscoelasticity
Completely monotone functions in fractional relaxation ...€¦ · This lecture is partly based on...
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Third Workshop on Fractional Calculus, Probability and Non-Local OperatorsFCPNLO 2015, Basque Center of Applied Mathematics,
Bilbao, 18�20 November 2015
Completely monotone functions
in fractional relaxation processes
Francesco MAINARDI
Department of Physics, University of BolognaVia Irnerio 46, I-40126 Bologna, [email protected]
http://www.fracalmo.org
BCAM: 19 November 2015
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Acknowledgements
This research has been carried out in the framework of the projectFractional Calculus Modelling, see http://www.fracalmo.org
This lecture is partly based on the author's books and papers- F. Mainardi: Fractional Calculus and Waves in Linear Viscoelasticity",Imperial College Press, London (2010), pp. 340, ISBN 978-1-84816-329-4.- R. Goren�o, A.A Kilbas, F. Mainardi and S.V. Rogosin: Mittag-Le�er
Functions. Related Topics and Applications, Springer, Berlin (2014), pp.420. ISBN 978-3-662-43929-6, Springer Monographs in Mathematics.- R. Goren�o and F. Mainardi: Fractional relaxation of distributed order,in M. Novak (Editor): Complexus Mundi: Emergent Patterns in Nature,World Scienti�c, Singapore (2006), pp. 33�42.- E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr: Models basedon Mittag-Le�er functions for anomalous relaxation in dielectrics. The
European Physical Journal, Special Topics, Vol. 193, pp. 161�171 (2011).Revised version as E-print http://arxiv.org/abs/1106.1761- F. Mainardi and R. Garrappa: On complete monotonicity of thePrabhakar function and non-Debye relaxation in dielectrics. Journal of
Computational Physics, Vol. 293 (2015), pp. 70�80.
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Abstract
In this talk the condition of complete monotonicity (CM) isdiscussed for the response functions characterizing relaxationprocesses modelled by constitutive equations of fractional order.Our models can concern both mechanical and dielectric relax-ation. Indeed, in view of the electro-mechanical analogy, linearviscoelastic and dielectric materials exhibit similar features asfar as the time-dependent response functions are concerned.We point out that CM is an essential property for thephysical acceptability and realizability of the models sinceit ensures, for instance, that in isolated systems the energydecays monotonically as expected from physical considerations.Studying the conditions under which the response function of asystem is CM is therefore of fundamental importance.The purpose of this talk is to summarize the results obtained bythe author with some collaborators for dielectric media wherethe response functions are shown to be of the Mittag-Le�ertype.
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Contents
1 A survey on completely monotoneand Bernstein functions 61.1 De�nitions . . . . . . . . . . . . . . . . . . . . . 61.2 Some basic criteria and examples . . . . . . . . . 11
2 The Mittag-Le�er functions 202.1 De�nitions . . . . . . . . . . . . . . . . . . . . . 202.2 The Laplace transform pairs . . . . . . . . . . . 212.3 Complete monotonicity . . . . . . . . . . . . . . 23
3 The dielectric relaxation phenomenon 403.1 Introduction to dielectric relaxation . . . . . . . 403.2 The Cole-Cole relaxation model . . . . . . . . . 443.3 The Davidson�Cole relaxation model . . . . . . . 473.4 The Havriliak�Negami relaxation model . . . . . 50
4 Bibliography 60
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5 Appendix A: The spectral distribution of thePrabahakar function 67
6 Appendix B: Formal demonstration of K1α,β(r) =
Kα,β(r) 69
7 Appendix C: Complete monotonicity fromLaplace transform 72
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1. A survey on completely monotone
and Bernstein functions
1.1. De�nitions
We recall that a function φ(t) de�ned for t ≥ 0 is said tobe completely monotone (CM ) if it is a non-negative within�nitely many derivatives that alternate in sign. In other words
(−1)ndn
dtnφ(t) ≥ 0 , n = 0, 1, 2, . . . , t ≥ 0 . (1.1)
A necessary and su�cient condition is given by Bernstein'stheorem, according to which a function φ(t) is CM in R+ ifand only if it is the restriction to the positive real semi-axis ofthe Laplace transform of a positive measure.
For sake of simplicity we agree to represent φ(t) as
φ(t) =
∫ ∞0
e−rtKφ(r) dr , Kφ(r) ≥ 0 , (1.2)
where Kφ(r) is a non-negative (ordinary or generalized)function.
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Alternatively, introducing τ = 1/r we can write Eq. (2) as
φ(t) =
∫ ∞0
e−t/τ Hφ(τ) dτ , Hφ(τ) ≥ 0 , (1.3)
whereHφ(τ) = Kφ(1/τ)/τ 2 , (1.4)
is a non-negative (ordinary or generalized) function alike Kφ(r).
Furthermore, a function ψ(t) de�ned for t ≥ 0 is said to be aBernstein function if it is non-negative and its �rst derivativeis a CM function. In other words
ψ(t) ≥ 0 , (−1)n−1 dn
dtnψ(t) ≥ 0 , n = 1, 2, . . . , t ≥ 0 . (1.5)
Thus the CM and Bernstein functions are special non-negativefunctions de�ned on the positive real semi-axis, the former beingdecreasing and convex, the latter increasing and concave.
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Relevant and simplest examples of CM and Bernstein functionsare φ(t) = e−t and ψ(t) = 1− e−t, respectively.
For more details on these classes of functions we refermainly to [Berg and Forst (1975), Gripenberg et al. (1990),Miller and Samko (2001), Schilling et al. (2012)], where thereader can �nd rigorous and exhaustive treatments. A morepractical approach to this matter can be found in [Feller (1971)].
We agree to represent a Bernstein function ψ(t) as
ψ(t) = a+ bt+
∫ ∞0
(1− e−rt
)Kψ(r) dr , a, b,Kψ(r) ≥ 0 ,
(1.6)where Kψ(r) is a non-negative (ordinary or generalized)function, and a = ψ(0+), b = lim
t→∞ψ(t)/t.
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Alternatively, introducing τ = 1/r, we can write Eq. (1.6) as
ψ(t) = a+ bt+
∫ ∞0
(1− e−t/τ
)Hψ(τ) dτ , Hψ(τ) ≥ 0 ,
(1.7)where
Hψ(τ) = Kψ(1/τ)/τ 2 , (1.8)
is a non-negative (ordinary or generalized) function alikeKψ(r).
We note that if for a certain α ∈ (0, 1), ψ(t) = O(tα) as t→∞,then ∫ ∞
0
Kψ(r) dr =
∫ ∞0
Kψ(τ) dτ =∞ . (1.9)
The functions K(r) and H(τ) are usually referred to as thefrequency and time spectral functions, respectively. Theyare interrelated through the di�erential form
K(r) dr = H(τ) dτ . (1.10)
The spectral functions are relevant to characterize the processesof relaxation and creep in linear viscoelastic and dielectricmaterials.
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The spectral functions can be uniquely determined from thederivative of the corresponding functions φ(t) and ψ(t) by usingthe following relations akin to Laplace transform pairs. We get∫ ∞
0
r Kφ(r) e−tr dr = − ddtφ(t) , (1.11)∫ ∞
0
r Kψ(r) e−tr dr =d
dtψ(t) + b , (1.12)
showing that−r Kφ(r) and r Kψ(r) can be viewed as the inverseLaplace transforms of φ̇(t) and ψ̇(t) + b, respectively, wheret is now considered the Laplace transform variable instead ofthe usual r and viceversa. The time spectral functions can bederived from the corresponding frequency spectral functions byEqs. (1.4), (1.8). Thus, adopting the connective symbol ÷ forLaplace transform pairs, where in the LHS we put the originalfunction and in the RHS its Laplace transform, we re-write Eqs.(1.11), (1.12) as
−r Kφ(r) ÷ φ̇(t) , (1.11′)
r Kψ(r) = ÷ ψ̇(t) + b . (1.12′)
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1.2. Some basic criteria and examples
The following elementary functions are CM
e−at , a ≥ 0 ,
1
λ+ µt, λ ≥ 0 , µ ≥ 0 , ν ≥ 0 , (1.13)
log(b+
c
t
), b ≥ 1 , c > 0 . (1.14)
A trivial observation is the φ(t) is CM, then φ(2m)(t) and−φ(2m+1) are also CM
Theorem 1. If φ1(t) and φ2(t) are CM, then aφ1(t) + bφ2(t),where a and b are non negative constants and φ1(t)φ2(t) arealso CM
The following elementary functions are Bernstein
tα , 0 < α ≤ 1 , (1.15)
log(1 + at) , a > 0 , (1.16)
(1 + at)α − 1
α, a > 0 , α ≤ 1 . (1.17)
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Remark on Eq (1.17): In the extended Je�rey-Lomnitz crreplaw the complete range α ≤ 1 (rather than 0 ≤ α ≤ 1) yields acontinuous transition from a Hooke elastic solid with no creep(α→−∞) to a Maxwell �uid with linear creep (α=1) passingthrough the Lomnitz viscoelastic body with logarithmic creep(α= 0), which separates solid-like from �uid-like behaviors.
In the expression for the extended Je�reys-Lomnitz creep law,it is convenient to separately consider four cases:
t ≥ 0 , ψ(t) =
t/τ0 , α = 1 ,
(1 + t/τ0)α − 1
α, 0 < α < 1 ,
log(1 + t/τ0) , α = 0 ,
1− (1 + t/τ0)−|α|
|α| α < 0 .
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The behaviour of ψ(t) as a function of the dimensionless timet/τ0 is illustrated in the Figures below, for some values of α inthe range −2 ≤ α ≤ 1, adopting a logarithmic time scale and alinear time scale.
.
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
0.0
0.5
1.0
1.5
2.0
ψ (
t)
−2 −1 0 1 2
LOG10(t/τ0)
α=1 0.5 0
−0.5
−1.0
−1.5
−2
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
0.0
0.5
1.0
1.5
2.0
ψ (
t)
0.0 0.5 1.0 1.5 2.0
t/τ0
α=1
0.5
0
−0.5
−1.0−1.5−2.0
F. Mainardi and G. Spada: On the viscoelastic characterizationof the Je�reys�Lomnitz law of creep, Rheol. Acta 51 (2012),783�791. [E-print: http://arxiv.org/abs/1112.5543]
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Theorem 2. Let φ(t) be CM and let ψ(t) be Bernstein, theφ[ψ(t)] also is CM function. A noteworthy example is φ(t) =exp[−ψ(t)].
Corollary 1. Let f(t) and φ(t) be CM Then
f
(a+ b
∫ t
0
φ(t′) dt′), a ≥ 0 , b ≥ 0 , (1.18)
also is CM
Corollary 2. Let f(t) be CM and f(0) < ∞. Then thefunctions
1
[A− f(t)]µ, A ≥ f(0) , µ ≥ 0 , (1.19)
and
− log
[1− f(t)
A
], A ≥ f(0) , (1.20)
are CM It also follows that the function
f ′(t)
A− f(t), A ≥ f(0) , (1.21)
is c.m since reduces to minus derivative of the function above.
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We note some particular cases of CM functions of the abovetypes.
exp (−atα) , a ≥ 0 , 0 ≤ α ≤ 1 , (1.22)
1
[a+ b log(1 + t)]µ, a ≥ 0 b ≥ 0 , µ ≥ 0 , (1.23)
1
(a− be−t)µ , a ≥ b > 0 , µ ≥ 0 , (1.24)
Another version of the statement for composite functions isgiven in terms of power series with non-negative coe�cients bythe following theorem.
Theorem 3. Let y = φ(t) be CM and let the power series
f(y) =∞∑n=0
anyn (1.25)
converge for all y in the range of the function y = φ(t). Ifan ≥ 0 for all n = 0, 1, 2, . . . . Then f [φ(t)] is CM
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Corollary 3. If φ(t) is CM, then exp[φ(t)] is CM. In particular,the functions
exp(atα) , a ≥ 0 , α ≤ 0 , (1.26)
(1 + t)a/t = exp[a log(1 + t)/t] , a ≥ 0 , (1.27)(a+
b
t
)µ=exp
{µ log
(a+
b
t
)}, a ≥ 1, b > 0, µ ≥ 0.
(1.28)Remark A natural idea is to pass from series representationwith positive coe�cients, as in Theorem 3, to integral trans-forms with non-negative densities. Let
f(t) =
∫ d
c
K(t, τ) g(τ) dτ, 0 ≤ c < d ≤ ∞. (1.29)
Obviously, if K(t, τ) is CM in t for all τ ∈ (0,∞) and g(τ) isnon-negative, the formal di�erentiation shows that f(t) also isCM. In view of this Remark [Miller and Samko (2001)] we mayextend the result (1.28) by proving the CM of(
a+b
tα
)µ, a ≥ 0, b > 0 µ ≥ 0, 0 ≤ α ≤ 1. (1.30)
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Furthermore we recall a noteworthy property relating CM andBernstein function in addition to their composition alreadydescribed in Theorem 2. If ψ(t) is a Bernstein function φ(t) =ψ(t)/t is CM.
By the way we could complement our analysis in notingthat there are two relevant subclasses for CM and Bernsteinfunctions: the Stieltjes functions and the CompleteBernstein functions, respectively well discussed in the book[Schilling et al. (2012)] to which we address the interestingreaders.
We prefer to conclude this introductory part with two theoremsrelating CM and Laplace transforms.
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There is a sort of inverse of the Bernstein theorem, according towhich the inverse Laplace transform of a CM function is non-negative and viceversa, under suitable regularity conditions:
φ(t) ≥ 0 , t ≥ 0 ⇐⇒ φ̃(s)CM function , s > 0 , (1.31)
where φ̃(s) is the Laplace transform of φ(t).
This property may be justi�ed by the previous Remark onintegral transforms (1.29) and by the Post-Widder formula forthe inversion of the Laplace transform:
φ(t) = limn→∞
(−1)n
n!
(nt
)n+1
φ̃(n)(nt
). (1.32)
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Herewith we report a theorem foundin [Gripenberg et al. (1990)], see Theorem 2.6, pp. 144-145,that provides necessary and su�cient conditions to ensure theCM of a locally integrable function f(t) in t ≥ 0 based on itsLaplace transform f̃(s).
Theorem The Laplace transform f̃(s) of a function f(t) thatis locally integrable on IR+ and CM has the following properties:(i) f̃(s) an analytical extension to the region C − IR−;(ii) f̃(s) = f̃ ∗(s) for s ∈ (0,∞);(iii) lim
s→∞f̃(s) = 0;
(iv) Im{f̃(s)} < 0 for Im{s} > 0;(v) Im{s f̃(s)} ≥ 0 for Im{s} > 0 and f̃(x) ≥ 0 for x ∈ (0,∞).
Conversely, every function f̃(s) that satis�es (i)�(iii) togetherwith (iv) or (v), is the Laplace transform of a function f(t),which is locally integrable on IR+ and CM on (0,∞).
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2. The Mittag-Le�er functions
2.1. De�nitions
The 3-parameter Mittag-Le�er function
Eγα,β(z) :=
∞∑n=0
(γ)nn!Γ(αn+ β)
zn , (2.1)
Re{α} > 0, Re{β} > 0, Re{γ} > 0 ,
(γ)n = γ(γ + 1) . . . (γ + n− 1) =Γ(γ + n)
Γ(γ).
For γ = 1 we recover the 2-parameter Mittag-Le�er function
Eα,β(z) :=∞∑n=0
zn
Γ(αn+ β), (2.2)
and for γ = β = 1 we recover the standard Mittag-Le�erfunction
Eα(z) :=∞∑n=0
zn
Γ(αn+ 1). (2.3)
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2.2. The Laplace transform pairs
Let us now consider the relevant formulas of Laplace transformpairs related to the above three functions, already known in theliterature for α, β, γ > 0 when the independent variable is realof type at where t > 0 is interpreted as time and a as a certainconstant.
Let us start with the most general function. Substituting theseries representation of the Prabhakar generalized Mittag-Le�erfunction in the Laplace transformation yields the identity∫ ∞
0
e−st tβ − 1Eγα,β(atα) dt=s−β
∞∑n=0
Γ(γ + n)
Γ(γ)
(as
). (2.4)
On the other hand (binomial series)
(1 + z)−γ =∞∑n=0
Γ(1− γ)
Γ(1− γ − n)n!zn =
∞∑n=0
(−1)nΓ(γ + n)
Γ(γ)n!zn .
(2.5)
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Comparison of Eq. (2.4) and Eq. (2.5) yields the Laplacetransform pair
tβ − 1Eγα,β (atα) ÷ s−β
(1− as−α)γ=
sαγ−β
(sα − a)γ. (2.6)
Eq. (2.6) holds (by analytic continuation) for Re(α) > 0,Re(β) > 0.
In particular we get the known Laplace transform pairs, see e.g.[Mainardi (2010), Podlubny (1999)],
tβ − 1Eα,β(atα) ÷ sα−β
sα − a =s−β
1− as−α , (2.7)
Eα(atα) ÷ sα−1
sα − a =s−1
1− as−α . (2.8)
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2.3. Complete monotonicity
We recall that a function f(t) with t ≥ 0 is completely monotone(CM) if it is positive and its derivatives are alternating in sign,
(−1)nf (n)(t) > 0 , t ≥ 0 . (2.9)
Thus, for the Bernstein theorem that states a necessary andsu�cient condition for the CM, the function can be expressed asa real Laplace transform of non-negative (generalized) function,
f(t) =
∫ ∞0
e−rt K(r) dr , K(r) ≥ 0 , t ≥ 0 . (2.10)
By the way, the determination of such non-negative functionK(r) (the Laplace measure) is a standard method to prove theCM of a given function de�ned in the positive real axis IR+. Inphysical applications the function K(r) is usually referred to asthe spectral distribution function, in that it is related to the factthat the process governed by f(t) can be expressed in terms of acontinuous distribution of elementary (exponential) relaxationprocesses with frequencies r on the whole range (0,∞).
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In the case of the pure exponential f(t) = exp(−λt) with agiven relaxation frequency λ > 0 we have K(r;λ) = δ(r − λ).
Since f̃(s) turns to be the iterated Laplace transform of K(r)we recognize that f̃(s) is the Stieltjes transform of K(r) andtherefore the spectral distribution can be determined as theinverse Stieltjes transform of f̃(s) via the Titchmarsh inversionformula, see e.g. [Titchmarsh (1937), Widder (1946)],
f̃(s) =
∫ ∞0
K(r
s+ rdr, K(r) = ∓1
πIm[f̃(s)
∣∣∣s=re±iπ
]. (2.11)
.
As a consequence, a method for proving CM suitable forour Mittag-Le�er functions (2.6) (2.7) (2.8) is that to arriveat the non-negative spectral distribution K(r) by invertingthe corresponding Laplace transform by using the Bromwichcontour integral. Indeed this was the method followed by[Goren�o and Mainardi (1997)] to prove the CM of Eα(−t−α)for 0 < α < 1 and to determine the corresponding spectralfunction.
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We recall the result
Eα(−tα)=
∫ ∞0
e−rtKα(r) dr, Kα(r)=−1
πIm
[sα−1
sα + 1
∣∣∣∣s=reiπ
],
(2.12)
Kα(r)=1
π r
sin(απ)
rα + 2 cos(απ) + r−α=
1
π
rα−1 sin (απ)
r2α + 2 rα cos (απ) + 1.
(2.13)As a matter of fact the functionKα(r) was derived as an exercisein complex analysis by evaluating the contribution on the branchcut (the negative real axis) of the Bromwich integral (bendingthe Bromwich path into the equivalent Hankel loop) and turnsout to be provided by the so-called Titchmarsh formula. Weeasily recognize
Kα(r) ≥ 0 if 0 < α ≤ 1 , (2.14)
including the limiting case α = 1 where our Mittag-le�erfunction reduces to the exponential exp(−t) and K1(r) =δ(r− 1). In fact, the denominator is non negative being greateror equal to (rα− 1)2 and the numerator is non-negative as soonas the sin function is non-negative.
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In Fig. 2.1 we show some plots of Ψ(t) = Eα(−tα) for somevalues of the parameter α. It is worth to note the di�erent ratesof decay of Eα(−tα) for small and large times. In fact the decayis very fast as t→ 0+ and very slow as t→ +∞.
.
Fig. 2.1 Plots of the Mittag-Le�er function Eα(−tα) for α =0.25, 0.50, 0.75, 1.; Left: in linear-linear scales; Right in log-logscales
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In Fig. 2.2 we show Kα(r) for some values of the parameter α.Of course for α = 1 the Mittag-Le�er function reduces to theexponential function exp(−t) and the corresponding spectraldistribution is the Dirac delta generalized function centred atr = 1, namely δ(r − 1).
0 0.5 1 1.5 20
0.5
1
Kα(r)
α = 0.25
α = 0.50
α = 0.75
α = 0.90
r
Fig. 2.2 Plots of the spectral function Kα(r) for α =0.25, 0.50, 0.75, 0.90 in the frequency range 0 ≤ r ≤ 2.
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As a matter of fact, Kα(r) provides an interesting spectralrepresentation of eα(t) in frequencies. With the changeof variable τ = 1/r we get the corresponding spectralrepresentation in relaxation times, namely
Eα(t) =
∫ ∞0
e−t/τHα(τ) dτ , Hα(τ) = τ−2Kα(1/τ) , (2.15)
that can be better interpreted as a continuous distributionsof elementary (i.e. exponential) relaxation processes. Asa consequence we get the identity between the two spectraldistributions, that is Kα(r) = Hα(τ), because we get
Hα(τ) =1
π
τα−1 sin (απ)
τ 2α + 2 τα cos (απ) + 1, (2.13′)
a surprising fact pointed out in Linear Viscoelasticity bythe author in his book [Mainardi (2010)]. This kind ofuniversal/scaling property seems a peculiar one for our Mittag-Le�er function Eα(−tα).
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Before deriving the conditions of CM and the correspondingspectral function for the Mittag-Le�er function in threeparameters in Eq.(2.6) let us revisit the conditions of CM for thefunction in two parameters in Eq.(2.7) following the approach byGoren�o and Mainardi [Goren�o and Mainardi (1997)]. Sincethe argument of our function atα must be negative we assumea = −1 (without loss of generality) so the correspondingLaplace transform pair reads from Eq.(2.7),
tβ−1Eα,β(−tα) ÷ s−β
1 + s−α=
sα−β
sα + 1. (2.16)
We prove the existence of the corresponding spectral distribu-tion using the complex Bromwich formula to invert the Laplacetransform.
Taking 0 < α < 1 the denominator does not exhibit any zeroso, bending the Bromwich path into the equivalent Hankel path(the well known loop around the negative real semi-axis), weget
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tβ−1Eα,β(−tα)=
∫ ∞0
e−rtKα,β(r) dr ,
Kα,β(r)=−1
πIm
[sα−β
sα + 1
∣∣∣∣s=reiπ
](2.17)
Kα,β(r)=rα−β
π
sin [(β − α)π] + rα sin (βπ)
r2α + 2rα cos(απ) + 1. (2.18)
We easily recognize
Kα,β(r) ≥ 0 if 0 < α ≤ β ≤ 1 , (2.19)
In fact, the denominator in Eq.(2.18) is non negative beinggreater or equal to (rα− 1)2 and the numerator is non negativeas soon as the two sin functions are both non-negative.
The particular cases β = 1 with 0 < α ≤ 1 are of courserecovered.
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We recall that for the Mittag-Le�er functions in one andtwo-order parameters Eα(z), Eα,β the conditions to beCM were proved respectively by Pollard [Pollard (1948)] in1948 and by Schneider [Schneider (1996)] in 1996, see also[Miller and Samko (1997), Miller and Samko (2001)] for furtherdetails. These conditions require the independent variable to bereal and negative (we write z = −x with x ≥ 0) and the order-parameters such that 0 < α ≤ 1 for Eα(−x), and 0 < α ≤ 1and β ≥ α for Eα,β(−x).
As a consequence, the CM property of Eα(−tα) can also beseen as a consequence of the result by Pollard because thetransformation x = tα is just a Bernstein function for α ∈ (0, 1].
Furthermore, we note that the conditions (2.19) on theparameters α and β can also be justi�ed by noting that in thiscase the function tβ−1Eα,β(−tα) is CM as a product of two CMfunctions. In fact tβ−1 is CM if β ≤ 1 whereas Eα,β(−tα) is CMif 0 < α ≤ 1 and β ≥ α.
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Fig. 2.3 Kα,β(r) calculated for α = 0.9.
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Fig. 2.4Kα,β(r) calculated for α = 0.75.
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Fig. 2.5 Kα,β(r) calculated for α = 0.5.
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Fig. 2.6 Kα,β(r) calculated for α = 0.25.
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We �nally devote our attention to the more general three-parameters function
ξG(t) := tβ − 1Eγα,β (−tα) , (2.20)
with Laplace transform (as derived from (2.7) with a = −1)
ξ̃G(s) =sαγ−β
(sα + 1)γ, (2.21)
where the notation ξG(t) been introduced for future conveniencein dielectric models.
In analogy with the previous computing, we get
ξG(t) =
∫ ∞0
e−rtKγα,β(r) dr ,
where the spectral distribution is to be computed by applyingthe Titchmarsh formula to the Laplace transform (2.21).
We get:
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Kγα,β(r) =
r−β
πIm
{eiβπ
(rα + e−iαπ
rα + 2 cos(απ) + r−α
)γ}= −r
αγ−β
πIm
{ei(αγ−β)π
(rαeiαπ + 1)γ
}
from which, after standard manipulations in complex analysis,we get
Kγα,β(r) =
rαγ−β
π
sin [γ θα(r) + (β − αγ)π]
(r2α + 2rα cos(απ) + 1)γ/2, (2.23)
where
θα(r) := arctan
[rα sin(πα)
rα cos(πα) + 1
]∈ [0, π] . (2.24)
For details we refer to the Appendix A where a warning on thecorrect branch of the function arctan is also outlined.
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We easily recognize that θα(r) is a non-negative and increasingfunction of r limited by απ ≤ π, as shown in the Fig. 2.7, wherethe dotted lines indicate the limit values απ. In fact for r � 1it is direct to check its asymptotic behaviour
rα sin(πα)
rα cos(πα) + 1=
sin(πα)
cos(πα) + 1/rα≤ sin(πα)
cos(πα)= tan(πα) .
10−4
10−3
10−2
10−1
100
101
102
103
104
0
0.5
1
1.5
2
2.5
r
θ α(r)
α = 0.25
α = 0.50
α = 0.75
Fig. 2.7The function θα(r) for α = 0.25, α = 0.50 and α =0.75.
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Then we recognize that in order the spectral distribution to benon-negative the argument of the sin function in the numeratormust be included in the closed interval [0, π] and henceforth we�nd the conditions
0 < α ≤ 1, 0 < αγ ≤ β ≤ 1. (2.25)
These conditions were formerly stated by the author incollaboration with the Brazilian colleagues Capelas and Vazin [Capelas, Mainardi and Vaz (2011)] by using a di�erentmethod. based on the requirements on the Laplace transformstated in the treatise [Gripenberg et al. (1990)], see Theorem2.6, pp. 144-145. Indeed these requirements provide necessaryand su�cient conditions to ensure the CM of a locally integrablefunction from its Laplace transform.
For this approach we refer the reader to Appendix B.
It is not straightforward to derive the spectral function (2.18) forγ = 1 from the the spectral function (2.23)-(2.24) with a genericγ 6= 1. For reader's convenience the derivation is reported inAppendix C.
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3. The dielectric relaxation phenomenon
3.1. Introduction to dielectric relaxation
We brie�y outline the theory of dielectric polarization inorder to introduce the mathematical models for anomalousrelaxation present in complex materials based on the Mittag-Le�er functions.
Under the in�uence of the electric �eld, the matter becomespolarized. For a perfect isotropic dielectric the interdependencebetween the electric �eld E and the polarization P is describedby the law:
P = ε0[(εs − ε∞)(ε̂(iω)− 1)]E = ε0[(εs − ε∞)χ̂(iω)]E , (3.1)
where εs and ε∞ are the static and in�nite dielectric constants,and ε̂(iω) and χ̂(iω) are respectively the normalized permit-tivity and susceptibility depending on the frequency. ω of theexternal �eld. They are speci�c characteristics of the medium,determined by matching experimental data into a theoreticalmodel.
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The classical Debye model has given an expression for thenormalized permittivity as
ε̂(iω) =1
1 + iωτ= ε′(ω)− iε′′(ω) , (3.2)
where τ is the only relaxation time expected. As a consequencethe response function ξ(t) (= inverse Laplace transform of thedielectric permittivity) is purely exponential.
The Debye model was the �rst relaxation law based on theStatistical Mechanics and formulated in terms of the Brownianmotion. However, it is useful in a little variety of cases. Indeed,in complex materials, the Debye phenomenology of relaxationprocesses breaks down and a power law decay is generallyfound. Furthermore it does not �t the observed distributionsof relaxation times.
Models for the non-Debye (= anomalous) relaxation are thusnecessary especially in complex materials like the biologicalones.
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The Laplace transform pairs for the Mittag-Le�er functions inone, two and three parameters with t ≥ 0 and s = iω discussedin the previous Section can be used as mathematical modesfor the response function ξ(t) and for the complex permittivityε̂(ω) to take into account the anomalous relaxation relaxationin dielectrics.
The corresponding spectral functions can be used to characterizethe models by means of suitable distributions of exponentialrelaxation processes.
Furthermore, it is known that the dielectric models are alsodistinguishable by inspection of their (so-called) Cole-Cole plotsthat exhibit the imaginary part versus real part of the complexpermittivity .
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We now show how the three classical models for anomalousrelaxation referred to Cole�Cole (C-C), Davidson�Cole (D-C)and Havriliak�Negami (H-N) are contained in our general modelwhen αγ = β (taking for simplicity τ = 1)
sαγ−β
(sα + 1)γ=⇒ 1
(sα + 1)γ
so that
αγ = β
0 < α < 1 , γ = 1 : C-C {α} ,α = 1 , 0 < γ < 1 : D-C {γ} ,0 < α < 1 , 0 < γ < 1 : H-N {α, γ} .
(3.3)
However we can generalize the H-N model extending its originalrange of γ to
1 < γ < 1/α,
keeping the complete monotonicity of the correspondingresponse function, see [Capelas, Mainardi and Vaz (2011)],[Mainardi and Garrappa (2015)].
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3.2. The Cole-Cole relaxation model
The C-C relaxation model is a non-Debye relaxation modeldepending on one parameter, say α (0 < α < 1), see[Cole and Cole (1941), Cole and Cole (1942)], that for α = 1reduces to the standard Debye model. We have for 0 < α < 1:
ξ̃C-C(s) =1
1 + sα÷ ξC-C(t) = tα−1E1
α,α(−tα) = − d
dtEα(−tα) .
(3.4)The spectral distribution is thus obtained from that of the two-parameter Mittag-Le�er function (2.18) for α = β and alsofrom that of Eα(−tα) by multiplying by r:
KC-C(r) := K1α,α(r)=Kα,α(r)=
1
π
rα sin(απ)
r2α + 2rα cos(απ) + 1.
(3.5)We note its power law decay for r →∞: KC-C(r) ∼ 1/rα, andits maximum value in r = 1
KC-C(r = 1) =1
2π
sinαπ
cosαπ + 1=
1
2πtan
απ
2. (3.6)
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Fig. 3.1 The spectral distribution for the Cole-Cole modelKC-C(r) := K1
α,α(r) calculated for α = 0.9, α = 0.75, α = 0.5,α = 0.25 .
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The Cole-Cole plot for this model is shown in Fig. 3.2 where theapexes of the arcs correspond to the mean relaxation frequency.
Fig 3.2 Cole-Cole plot for Cole-Cole and Debye models
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3.3. The Davidson�Cole relaxation model
The D-C relaxation model is a non-Debye relaxation modeldepending on one parameter, say γ (0 < γ < 1), see[Davidson and Cole], that for γ = 1 reduces to the standardDebye model. The corresponding complex susceptibility (s =iω) and response function read for 0 < γ < 1:
ξ̃D-C(s) =1
(1 + s)γ÷ ξD-C(t) = tγ−1Eγ
1,γ(−t) =tγ−1
Γ(γ)e−t , , .
(3.7)The spectral distribution for the Cole-Davidson model is easilyobtained
KD-C(r) := Kγ1,γ(r) =
0 , 0 < r < 1 ,
(r − 1)−γsin(γπ)
π, r > 1 ,
(3.8)
where we have used the identity Γ(γ)Γ(1− γ) =π
sin(γπ).
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Fig. 3.3 The spectral distribution for the Cole-Davidson model,KD-C(r) := Kγ
1,γ(r) calculated for γ = 0.9, γ = 0.75, γ = 0.5,γ = 0.25 .
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In Fig. 3.4 the Cole-Cole plots are shown for this model. Wenote skewed arcs where the maximum deviates from ω = 1.
3.4 Cole-Cole plot for Davidson-Cole model
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3.4. The Havriliak�Negami relaxation model
The clasical H-N relaxation model is a non-Debye relaxationmodel depending on two parameters, say α (0 < α < 1)and γ (0 < γ < 1), see [Havriliak and Negami (1966),Havriliak and Negami (1967)], that for α = γ = 1 reduces tothe standard Debye model.
It is worth to note that we have been able to extend the validityof the classical Havriliak-Negami model for 1 < γ < 1/α forany α ∈ (0, 1) because the CM of the response function is stillvalid.
The corresponding complex permittivity and response functionread
ξ̃H-N(s) =1
(1 + sα)γ÷ ξH-N(t) = tαγ−1Eγ
α,αγ(−tα) . (3.9)
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We recognize that the H-N relaxation model for γ = 1 and0 < α < 1 reduces to the C-C model, while for α = 1 andγ < 1 to the D-C model.
We also note that whereas for the C-C and H-N models (with0 < α < 1) the corresponding response functions decay like acertain negative power of time, for the D-C model, being α = 1the response function exhibits an exponential decay.
In the following slides we show the spectral functions and theCole-Cole plots for the H-N model with α = 0.50, 0.75 both inthe range 0 < γ < 1 and 1 ≤ γ < 1/α.
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0 1 2 3 4 50
0.05
0.1
0.15
0.2
r
KH−N(r)
γ = 0.25γ = 0.50γ = 0.75γ = 0.90
α = 0.50 0 < γ < 1 β = αγ
Fig. 3.5 The spectral distribution for the Havriliak-Negamimodel KH-N(r) for α = 0.5 and 0 < γ < 1.
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0 1 2 3 4 50
0.05
0.1
0.15
0.2
r
KH−N(r)
γ = 1.00
γ = 1.25
γ = 1.50
γ = 1.75
α = 0.50 1 ≤ γ < 1/α β = αγ
Fig. 3.6 The spectral distribution for the Havriliak-Negamimodel KH-N(r), for α = 0.5, and 1 ≤ γ < 2 = 1/α.
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0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
r
KH−N(r)
γ = 0.25γ = 0.50
γ = 0.75γ = 0.90
α = 0.75 0 < γ < 1 β = αγ
Fig. 3.7 The spectral distribution for the Havriliak-Negamimodel, KH-N(r), for α = 0.75 and 0 < γ < 1.
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0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
r
KH−N(r)
γ = 1.00
γ = 1.10
γ = 1.20
γ = 1.30
α = 0.75 1 ≤ γ < 1/α β = αγ
Fig. 3.8 The spectral distribution for the Havriliak-Negamimodel KH-N(r), for α = 0.75 and 1 ≤ γ < 4/3 = 1/α.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ε’
ε"
HN − α =0.5
γ =0.25 γ =0.5 γ =0.75 γ =0.9
Fig. 3.9 The Cole-Cole plot for the Havriliak-Negami modelfor α = 0.50 and 0 < γ < 1.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ε’
ε"
HN − α =0.5
γ =1 γ =1.25 γ =1.5 γ =1.75
Fig. 3.10 The Cole-Cole plot for the Havriliak-Negami modelfor α = 0.50 and 1 ≤ γ < 1/α.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ε’
ε"
HN − α =0.75
γ =0.25 γ =0.5 γ =0.75 γ =0.9
Fig. 3.11 The Cole-Cole plot for the Havriliak-Negami modelfor α = 0.75 and 0 < γ < 1.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ε’
ε"
HN − α =0.75
γ =1 γ =1.15 γ =1.2 γ =1.32
Fig. 3.12 The Cole-Cole plot for the Havriliak-Negami modelfor α = 0.75 and 1 ≤ γ < 1/α.
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4. Bibliography
References
[Berg and Forst (1975)] C. Berg and G. Forst, Potential Theory on Locally
Compact Abelian Groups, Springer, Berlin, 1975.
[Capelas, Mainardi and Vaz (2011)] E. Capelas de Oliveira, F. Mainardi and J.Vaz Jr, Models based on Mittag-Le�er functions for anomalous relaxationin dielectrics. The European Physical Journal, Special Topics 193, 161�171(2011). Revised version as E-print http://arxiv.org/abs/1106.1761
[Capelas, Mainardi and Vaz (2014)] E. Capelas de Oliveira, F. Mainardi and J.Vaz Jr, Fractional models of anomalous relaxation based on the Kilbas andSaigo function, Meccanica 49, 2049�2060 (2014).
[Capelas and Rosa (2015)] E. Capelas de Oliveira and E.C.F.A. Rosa, Relax-ation equations: fractional models, J. Phys. Math. 6 No 2, 1000146/1-7(2015). [E-print http://arxiv.org/abs/1510.01681v1]
[Cole and Cole (1941)] K.S. Cole and R.H. Cole, Dispersion and absorption indielectrics. I. Alternating current characteristics, J. Chem. Phys. 9, (1941)341�351.
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[Cole and Cole (1942)] K.S. Cole and R.H. Cole, Dispersion and absorption indielectrics. II. Direct current characteristics. J. Chem. Phys. 10, (1942)98�105.
[Davidson and Cole] D.W. Davidson and R.H. Cole, Dielectric relaxation inglycerol, propylene glycol, and n-propanol, J. Chem. Phys. 19, (1951)1484�1490.
[Diethelm (2010)] K. Diethelm, The Analysis of Fractional Di�erential Equa-
tions. An Application-Oriented Exposition Using Di�erential Operators
of Caputo Type. Springer, Berlin, 2010. [Springer Lecture Notes inMathematics No 2004]
[Feller (1971)] W. Feller, An Introduction to Probability Theory and its Appli-
cations, Vol. 2, 2-nd Edition, Wiley, New York, 1971.
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[Goren�o et al. (2013)] R. Goren�o , Yu. Luchko and M. Stojanovic, Fundamen-tal solution of a distributed order time-fractional di�usion-wave equationas probability density. Fract. Calc. Appl. Anal. 16 No 2, 297�316 (2013).
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[Goren�o and Mainardi (1997)] R. Goren�o and F. Mainardi, Fractional cal-culus: integral and di�erential equations of fractional order, in: A.Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus
in Continuum Mechanics, Springer Verlag, Wien, 1997, pp. 223�276. [E-print: http://arxiv.org/abs/0805.3823].
[Goren�o and Mainardi (2006)] - R. Goren�o and F. Mainardi: Fractionalrelaxation of distributed order, in M. Novak (Editor): Complexus Mundi:
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[Hanyga and Seredynska (2008)] A. Hanyga and M. Seredy«ska, On a mathe-matical framework for the constitutive equations of anisotropic dielectricrelaxation. J. Stat. Phys 131, (2008) 269�303.
[Havriliak and Negami (1966)] S. Havriliak Jr. and S. Negami, A complex planeanalysis of α-dispersions in some polymer systems, J. Polymer Sci. C 14,(1966) 99�117.
[Havriliak and Negami (1967)] S. Havriliak and S. Negami, A complex planerepresentation of dielectric and mechanical relaxation processes in somepolymers, Polymer 8 No 4, 161�210 (1967).
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[Mainardi (2010)] F. Mainardi, Fractional Calculus and Waves in Linear
Viscoelasticity, Imperial College Press, London, 2010.
[Mainardi (2014)] F. Mainardi, On some properties of the Mittag-Le�erfunction Eα(−tα), completely monotone for t > 0 with 0 < α < 1, Discreteand Continuous Dynamical Systems, Ser. B 19, No 7, 2267�2278 (2014).[E-print http://arxiv.org/abs/1305.0161]
[Mainardi and Garrappa (2015)] F. Mainardi and R. Garrappa, On completemonotonicity of the Prabhakar function and non-Debye relaxation indielectrics, J. Comput. Phys. 293, 70�80 (2015).
[Mainardi and Goren�o (2007)] F. Mainardi and R. Goren�o, Time-fractional derivatives in relaxation processes: a tutorial survey,Fract. Calc. Appl. Anal. 10 No 3, 269�308 (2007). [E-printhttp://arxiv.org/abs/0801.4914]
[Mainardi et al. (2007)] F. Mainardi, A. Mura, R. Goren�o and M. Sto-janovic, The two forms of fractional relaxation of distributed or-der, J. Vibration and Control 13, 1249�1268 (2007). [E-printhttp://arxiv.org/abs/cond-mat/0701131
[Mainardi et al. (2008)] F. Mainardi, A. Mura, G. Pagnini and R. Goren�o,Time-fractional di�usion of distributed order, J. Vibration and Control 14,1267�1290 (2008). [E-print http://arxiv.org/abs/cond-mat/0701132
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[Miller and Samko (1997)] K.S. Miller and S.G. Samko, A note on the completemonotonicity of the generalized Mittag-Le�er function. Real Anal.
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[Polito and Tomovski (2015)] F. Polito and Z. Tomovski, Some properties ofPrabhakar-type operators, E-print arXiv:1508.03224v2 [math.PR], 8Sept 2015, pp. 18.
[Pollard (1948)] H. Pollard, The completely monotonic character of the Mittag-Le�er function Eα(−x), Bull. Amer. Math. Soc. 54, 1115�1116 (1948).
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[Prabhakar (1971)] T.R. Prabhakar, A singular integral equation with ageneralized Mittag-Le�er function in the kernel, Yokohama Math. J. 19,7�15 (1971).
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[Zemanian (1972)] A.H. Zemanian, Realizability Theory for Continuous Linear
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5. Appendix A: The spectral distribution of
the Prabahakar function
For ease of presentation we have collected in this Appendix some detailsregarding the derivation of the spectral distribution Kγ
α,β(r).After applying the Titchmarsh inversion formula, from (2.?) we have
Kγα,β(r) =
r−β
πIm
{eiβπ
(rα + e−iαπ
rα + 2 cos(απ) + r−α
)γ}= −r
αγ−β
πIm
{ei(αγ−β)π
(rαeiαπ + 1)γ
}
= −rαγ−β
πIm
{ei(αγ−β)π
(rαeiαπ + 1)γ(rαe−iαπ + 1)γ
(rαe−iαπ + 1)γ
} (A.1)
It is now easy to check that the denominator is real and non-negative, so weset
ξ := (rαeiαπ + 1)(rαe−iαπ + 1) = r2α + 2rα cos(απ) + 1 , (A.2)
with ξ ≥ (rα − 1)2 ≥ 0 and consequently ξγ/2 ≥ 0. For the numerator we set
z := (rαe−iαπ + 1) = [rα cos(απ) + 1]− i rα sin(απ) = ρ e−iθ , (A.3)
where 0 ≤ θ ≤ π (being 0 < α ≤ 1). Then
ρ = |z| =√
[Re(z)]2 + [Im(z)]2 = ξ12 (A.4)
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and
tan θ = − Im(z)
Re(z)=
rα sin(πα)
rα cos(πα) + 1. (A.5)
Using the de Moivre's formula (cosψ + i sinψ)n = cos(nψ) + i sin(nψ) weget
Kγα,β(r) =−r
αγ−β
π
Im {[cos(αγ − β)π + i sin(αγ − β)π] [cos(γθ)− i sin(γθ)]}ξγ/2
=−rαγ−β
π
[− cos(αγ − β)π sin(γθ) + sin(αγ − β)π cos(γθ)]
ξγ/2
=−rαγ−β
π
sin [(αγ − β)π − γθ]ξγ/2
=rαγ−β
π
sin [γθ + (β − αγ)π]
ξγ/2
(A.6)where
θ = θα(r) := arctan
[rα sin(πα)
rα cos(πα) + 1
]∈ [0, π]. (A.7)
As noted by Zorn for the Havriliak-Negami model [Zorn (1999)], see our analysis
for αγ − β = 0, we need to chose the arctangent's value in the interval [0, π],
which is possible if one considers arctan(x) to be a multivalued function. In this
sense our proposed formula is always valid if only correctly interpreted. Staying
with the usual de�nition of arctan(x) as a function into [−π/2, π/2], one has to
add π to avoid the negative values instead of the changing of sign.
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6. Appendix B: Formal demonstration of
K1α,β(r) = Kα,β(r)
In this appendix we show how, for γ = 1, the spectral density of 3-parametersMittag-Le�er function Kγ
α,β(r) reduces to the 2-parameters spectral densityKα,β(r). Recalling Eqs. (2.23)-(2.24), for γ = 1 we have:
K1α,β(r) =
1
π
rα−β
(r2α + 2rα cos(απ) + 1)γ2
sin
[arctan
(rα sin(πα)
rα cos(πα) + 1
)+ (β − α)π
]=
1
π
rα−β
ξ1/2sin[θ + (β − α)π] ,
where we call ξ = r2α + 2rα cos(απ) + 1, and θ = arctan(
rα sin(πα)rα cos(πα)+1
)as well
as in Appendix A, just for brevity.
Recalling the trigonometric formula
sin(α+ β) = sin(α) cos(β) + cos(α) sin(β) ,
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we have:
K1α,β(r) =
1
π
rα−β
ξ12
[sin θ cos(β − α)π + cos θ sin(β − α)π
]=
1
π
rα−β
ξ12
cos θ[
tan θ cos(β − α)π + sin(β − α)π]
=1
π
rα−β
ξ12
cos θ
[rα sin(πα)
rα cos(πα) + 1cos(β − α)π + sin(β − α)π
].
Noting that
tan2 θ =
(rα sin(πα)
rα cos(πα) + 1
)2
=sin2 θ
cos2 θ=
1− cos2 θ
cos2 θ=
1
cos2 θ− 1
we can write:
cos2 θ =1
tan2 θ + 1=
(rα cos(πα) + 1)2
r2α(sin2(πα) + cos2(πα)
)+ 2rα cos(πα) + 1
=(rα cos(πα) + 1)2
ξ.
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Extracting the square root we �nd:
K1α,β(r) =
1
π
rα−β
ξ12
(rα cos(πα) + 1
ξ12
)[rα sin(πα)
rα cos(πα) + 1cos(β − α)π + sin(β − α)π
]=
1
π
rα−β
ξ
[rα sin(πα) cos(β − α)π + rα cos(πα) sin(β − α)π + sin(β − α)π
]=
1
π
rα−β
ξ
[rα sin(βπ) + sin(β − α)π
]=rα−β
π
sin [(β − α)π] + rα sin (βπ)
r2α + 2rα cos(απ) + 1= Kα,β(r)
quod erat demonstrandum.
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7. Appendix C: Complete monotonicity from
Laplace transform
Herewith we report a theorem found in [Gripenberg et al. (1990)], see Theorem2.6, pp. 144-145, that provides necessary and su�cient conditions to ensure theCM of a locally integrable function f(t) in t ≥ 0 based on its Laplace transformf̃(s).
This theorem was used by [Capelas, Mainardi and Vaz (2011)] in order toprovide a proof for the inequalities ensuring the CM of the function in the LHSof the Laplace transform pair
ξG(t) := tβ − 1Eγα,β (−tα) ÷ ξ̃G(s) =sαγ−β
(sα + 1)γ. (2.6)
that is the conditions ensuring the non-negativity of the corresponding spectralfunction Kγ
α,β(r) for r ≥ 0.
We recall these conditions consisting in the following inequalities that we like towrite in two equivalent forms,
0 < α ≤ 1, 0 < β ≤ 1, 0 < γ ≤ β
α⇐⇒ 0 < α ≤ 1, 0 < αγ ≤ β ≤ 1, (C.1)
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Hereafter, we recall this theorem from Section 1.
Theorem The Laplace transform f̃(s) of a function f(t) thatis locally integrable on IR+ and CM has the following properties:(i) f̃(s) an analytical extension to the region C − IR−;(ii) f̃(s) = f̃ ∗(s) for s ∈ (0,∞);(iii) lim
s→∞f̃(s) = 0;
(iv) Im{f̃(s)} < 0 for Im{s} > 0;(v) Im{s f̃(s)} ≥ 0 for Im{s} > 0 and f̃(x) ≥ 0 for x ∈ (0,∞).
Conversely, every function f̃(s) that satis�es (i)�(iii) togetherwith (iv) or (v), is the Laplace transform of a function f(t),which is locally integrable on IR+ and CM on (0,∞).
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We recognize that the requirements (i)�(iii) for ξ̃G(s) are surely satis�ed withthe �rst two conditions in Eq.(C.1), that is 0 < α < 1, 0 < β < 1 but for anyγ > 0. So for us it su�ces to determine which additional condition is impliedfrom the requirement (iv).
We will prove that this relevant condition is just 0 < αγ − β ≤ 0, namely0 < γ ≤ β/α, as stated in Eq.(C.1).
Recalling the expression of the Laplace transform in Eq.(2.6), the requirement(iv) reads:
Λ(s) := Im
[sαγ−β
(sα + 1)γ
]< 0 where Im{s} > 0 . (C.2)
Setting s = reiφ in the complex upper half-plane (Im{s} > 0) we consider
Λ(r, φ) := Im
[(reiφ)αγ−β(1 + rαe−iαφ)γ
|1 + rαeiαφ|2γ
]with r > 0, 0 < φ < π. (C.3)
To prove that Λ(r, φ) is negative it is su�cient to consider the numerator becausethe denominator is always non-negative. Setting
z = (reiφ)αγ−β (1 + rαe−iαφ)γ = ρ eiΨ , (C.4)
we must verify that the conditions on {α, β, γ} stated in Eq.(C.1) ensure that zhas negative imaginary part so it is located in the lower half plane with
−π < Ψ < 0 . (C.5)
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Let
z1 = rαγ−β ei(αγ−β)φ = ρ1 eiΨ1 , ρ1 = rαγ−β, Ψ1 = (αγ − β)φ , (C.6)
z2 = rα e−iαφ = ρ2 eiΨ2 , ρ2 = rα, Ψ2 = −αφ , (C.7)
and
z3 = (1 + z2)γ = ρ3 eiΨ3 , ρ3 = |1 + rα e−iαφ|γ , −αγφ < Ψ3 < 0 , (C.8)
so we can write the complex number in Eq.(C.4) as
z = z1 · z3 = ρ1 eiΨ1ρ3 eiΨ3 = ρ eiΨ with ρ = ρ1ρ3, Ψ = Ψ1 + Ψ3 . (C.9)
Now assuming 0 < φ < π a we �nd for αγ − β < 0:
−(β − αγ)π < Ψ1 < 0 , (C.10)
−αγπ < Ψ3 < 0 . (C.11)
For αγ − β = 0 we �nd Ψ1 = 0 and −αγπ = −βπ < Ψ3 < 0.
As a consequence for αγ − β ≤ 0, by summing (Ψ = Ψ1 + Ψ3), we �nally get
−π < −βπ < Ψ < 0 , (C.12)
so the inequality (C.5) is proved since 0 < β < 1. The limiting case β = 1 can
be seen to be included.
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