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Tema
Tendências em Matemática Aplicada e Computacional, 20, N. 2 (2019), 217-228
© 2019 Sociedade Brasileira de Matemática Aplicada e Computacional
www.scielo.br/tema
doi: 10.5540/tema.2019.020.02.0217

Integral Representations of Mittag-Leffler Function
on the Positive Real Axis

E.C. GRIGOLETTO1*, E.C. OLIVEIRA2 and
R.F. CAMARGO3

Received on August 31, 2018 / Accepted on January 14, 2019

ABSTRACT. We use the method for finding inverse Laplace transform without using integration on the
complex plane to show that the three-parameter Mittag-Leffler function, which appear in many problems
associated with fractional calculus, has similar integral representations on the positive real axis. Some of
them are presented.

Keywords: inverse Laplace transform, Mittag-Leffler functions, integral representations, fractional
calculus.

INTRODUCTION

The Mittag-Leffler function, introduced in 1902 by Gösta Mittag-Leffler [23], is important
in many fields, including description of the anomalous dielectric properties, probability the-
ory, statistics, viscoelasticity, random walks and dynamical systems [9, 10, 11, 14, 19, 25, 26].
Successively, generalizations of Mittag-Leffler function were proposed [27]. These functions
play a fundamental role in arbitrary order calculus, popularly known as fractional calculus
[4, 13, 18, 20, 22, 29], as well as the exponential function play in integer order calculus.

The classical Laplace transform is one of the most widely tools used in the literature for solving
integral equations and ordinary or partial differential equations, involving integer or fractional
order derivatives [1,8,31,33]. It is also used in many others applications such as electrical circuit
and signal processing [15, 21, 35]. In general, the Laplace inversion is done numerically due to
the impossibility of the exact inversion by means of an integration on the complex plane [7, 32].

*Corresponding author: Eliana Contharteze Grigoletto – E-mail: eliana.contharteze@unesp.br – https://orcid.org/
0000-0003-4336-5387
1Universidade do Estado de São Paulo (UNESP), Departamento de Bioprocessos e Biotecnologia - FCA, Campus
Botucatu, SP, Brazil E-mail: eliana.contharteze@unesp.br
2Universidade de Campinas, IMECC, Departamento de Matemática Aplicada, Campinas, SP, Brazil E-mail:
capelas@ime.unicamp.br
3Universidade do Estado de São Paulo (UNESP), Departamento de Matemática - FC, Campus Bauru, SP, Brazil E-mail:
rubens.camargo@unesp.br

https://orcid.org/0000-0003-4336-5387
https://orcid.org/0000-0003-4336-5387


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218 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

M. N. Berberan-Santos [2] proposed a new methodology for evaluation of the numerical in-
verse Laplace transform, without using integration on the complex plane, which was published
in 2005, and its methodology was used recently, for instance, to discuss the luminescence decay
of inorganic solids, and to obtain an integral representation of Mittag-Leffler relaxation function,
a special one-parameter Mittag-Leffler function [3]. Recently, the method to evaluate the inverse
Laplace transform without using integration on the complex plane was applied in [6] to find
integral representations on the positive real axis for some functions.

In this paper, with the method for finding inverse Laplace transform without using integration
on the complex plane we show that the three-parameter Mittag-Leffler function has integral
representations on the positive real axis.

The paper is organized as follows: in Section 1, we present some preliminaries concepts and the
methodology of inversion of the Laplace transform. In Section 2, using this methodology, we
express the integral representations of three-parameter Mittag-Leffler function and we use the
results from this study to discuss, in Section 3, a class of improper integrals, expressing them in
terms of the Mittag-Leffler functions. Concluding remarks close the paper.

1 PRELIMINARIES

In this section, we present the definition and some special cases of the Mittag-Leffler func-
tions, and a review of the methodology of inversion of the Laplace transform proposed by M.
N. Berberan-Santos in the following subsections.

1.1 Mittag-Leffler functions

The three-parameter Mittag-Leffler function, introduced by Prabhakar [27], of complex variable
z ∈ C, with complex parameters α, β , γ ∈ C, is defined by1

Eγ

α,β (z) =
∞

∑
j=0

(γ) j z j

Γ(α j+β ) j!
, (1.1)

with R(α)> 0, R(β )> 0 and R(γ)> 0, where

Γ(ρ) =
∫

∞

0
tρ−1e−tdt, (1.2)

is the Gamma function, and

(γ) j :=
Γ(γ + j)

Γ(γ)
, (1.3)

is the Pochhammer symbol. Taking γ = 1 in equation (1.1), we get the two-parameter Mittag-
Leffler function:

Eα,β (z) =
∞

∑
j=0

z j

Γ(α j+β )
. (1.4)

1R [ξ ] indicates the real part of ξ .

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GRIGOLETTO, OLIVEIRA and CAMARGO 219

When β = 1 in equation (1.4), we get the standard Mittag-Leffler function [23, 34]:

Eα(z) =
∞

∑
j=0

z j

Γ(α j+1)
. (1.5)

A function f ∈ C∞ (I) is to be completely monotonic (CM) on interval I if (−1)k f k(x) ≥ 0
for x ∈ I and k ∈ N = {0,1,2,3, . . .} [33]. Capelas et al. [5] showed that the function ϕ(t) =
tβ−1Eγ

α,β (−tα) is CM for 0 < αγ ≤ β ≤ 1. Particularly, these functions play an important rule
in anomalous dielectric relaxation where the memory effect appears specifically in the Havriliak-
Negami model, which contains Davidson-Cole model, Cole-Cole model and the classical Debye
model, as particular cases [17, 24].

An interesting functional relation case and the more simple special relations involving the Mittag-
Leffler functions Eα,β (z) and Eα(z), with z ∈ C, are given by the following equations:

E2α(z2) =
1
2
[Eα(z)+Eα(−z)] . (1.6)

E2,2(z) =
sinh
√

z√
z

. (1.7)

E1,2(z) =
ez−1

z
. (1.8)

E2(−z2) = cos z. (1.9)

E2(z2) = cosh z. (1.10)

1.2 Inversion of the Laplace transform

Let f (t) be a real function of (time) variable t ≥ 0. The Laplace transform of f , denoted by
F(s) = L [ f ] (s), is defined as follows:

L [ f ] (s) = F(s) =
∫

∞

0
e−st f (t)dt, (1.11)

whenever the integral converges for2 R [s]≥σ > 0, where s=σ + iτ , with σ and τ real numbers,
and F(s) = 0 for σ < 0. By means of equation (1.11) with Euler formula, we observe that

R [F(σ + iτ)] =
∫

∞

0
e−σt f (t) cos(tτ)dt (1.12)

and
I [F(σ + iτ)] =−

∫
∞

0
e−σt f (t) sin(tτ)dt. (1.13)

The expression for evaluation of the inverse Laplace transform of F(s), proposed by M. N.
Berberan-Santos [2], is given by

2R [s] indicates the real part of s and the imaginary part is denoted by I [s].

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220 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

f (t) =
eσt

π

∫
∞

0
[R [F(σ + iτ)]cos(tτ)−I [F(σ + iτ)]sin(tτ)]dτ, (1.14)

for t > 0 and any real number σ satisfying the condition σ ≥σ0 > 0, where σ0 is large enough that
F(s) is defined for R [s] ≥ σ0 > 0. The expression in equation (1.14) recovers the real function
whose Laplace transform is known.

From equation (1.14), for t > 0, we can write

π

2
[
e−σt f (t)+ eσt f (−t)

]
=
∫

∞

0
R [F(σ + iτ)] cos(tτ)dτ. (1.15)

The function f is such that f (ξ ) = 0 for ξ < 0. Since t > 0, equation (1.15) yields

f (t) =
2eσt

π

∫
∞

0
R [F(σ + iτ)]cos(tτ)dτ, (1.16)

Furthermore, in a similar way,

f (t) =−2eσt

π

∫
∞

0
I [F(σ + iτ)]sin(tτ)dτ. (1.17)

Namely, there are three possible cases to find the inverse Laplace transform of a function F(s),
they are given by equations (1.14), (1.16) and (1.17).

In this point, it is important to consider a simple example, illustrating the methodology that will
be used in this work: The Laplace transform of the exponential function is given by F(s) = 1

s−1 ,
for R [s]> 0. Choosing R [s] = σ = 2 and writing s = 2+ iτ , we have that R [F(2+ iτ)] = 1

1+τ2 .
Thus, from equation (1.16), we obtain

π

2
e−t =

∫
∞

0

cos(tτ)
1+ τ2 dτ, for t > 0. (1.18)

2 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

Some integral representations associated with the one-parameter Mittag-Leffler function can
be found in the following papers: [3, 12, 16]. Here we present integral representations for the
three-parameter Mittag-Leffler function and to prove the representations, we use the relations in
equations (1.14), (1.16) and (1.17). It is worthwhile to mention that the detail treatment of the
similar study can be found in [28, 30].

Theorem 1. Let α > 0, β > 0, γ > 0 and λ ∈R. Then, for t > 0, the three-parameter Mittag-Leffler
function Ψ(t) = Eγ

α,β (λ tα) has the following integral representations on the positive real axis

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GRIGOLETTO, OLIVEIRA and CAMARGO 221

Eγ

α,β (λ tα) =
t1−β eσ t

π

∫
∞

0

rαγ−β

r̃
cos
[
θ (αγ−β )− θ̃ + tτ

]
dτ, (2.1)

=
2 t1−β eσ t

π

∫
∞

0

rαγ−β

r̃
cos
[
θ (αγ−β )− θ̃

]
cos(tτ)dτ, (2.2)

= −2 t1−β eσ t

π

∫
∞

0

rαγ−β

r̃
sin
[
θ (αγ−β )− θ̃

]
sin(tτ)dτ, (2.3)

where σ > σ0 and σ0, r, θ , θ̃ and r̃ are defined by equations:

σ0 = |λ |
1
α . (2.4)

r cos θ = σ and r sin θ = τ. (2.5)

r̃
1
γ cos

(
θ̃

γ

)
= rα cos(θα)−λ and r̃

1
γ sin

(
θ̃

γ

)
= rα sin(θα) . (2.6)

Proof. The Laplace transform of the three-parameter Mittag-Leffler type function f (t) =

tβ−1Eγ

α,β (λ tα) is given by

L
[
tβ−1Eγ

α,β (λ tα)
]
(s) =

sαγ−β

(sα −λ )γ = F(s), for |λ s−α |< 1. (2.7)

The complex parameter s can be written as

s = σ + iτ = reiθ , (2.8)

with σ , τ ∈R, r > 0 and 0≤ θ ≤ 2π . In this way, from equation (2.8), we get equations in (2.5).

Expression (sα −λ )γ in the denominator of F(s) can be written in the following form:

(sα −λ )γ = r̃eiθ̃ . (2.9)

Replacing s by reiθ in equation (2.9), we get(
rαeiθα −λ

)γ

= r̃eiθ̃ ,

that is,

rαeiθα −λ = r̃
1
γ ei θ̃

γ . (2.10)

Separating real part and imaginary part in equation (2.10), we obtain the expressions in equation
(2.6).

We can thus conclude that

F(s) =
sαγ−β

(sα −λ )γ
=

(
reiθ

)αγ−β

r̃eiθ̃
=

rαγ−β

r̃
ei[θ(αγ−β )−θ̃ ].

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222 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

Through manipulation of F(s), we can separate its real and imaginary parts:

R [F(σ + iτ)] =
rαγ−β

r̃
cos
[
θ (αγ−β )− θ̃

]
(2.11)

and

I [F(σ + iτ)] =
rαγ−β

r̃
sin
[
θ (αγ−β )− θ̃

]
. (2.12)

Substituting equations (2.11) and (2.12) into equations (1.14), (1.16) and (1.17), we arrive at
equations (2.1), (2.2) and (2.3), respectively; and finally if we choose σ > σ0 = |λ |

1
α , then the

inequality |λ s−α |< 1 is satisfied. �

According to Theorem 1, the Mittag-Leffler function has similar integral representations, as we
have seen in the equations (2.1)-(2.3). We present some applications of this theorem in the next
section.

3 EVALUATION OF A CLASS OF IMPROPER INTEGRALS

In what follows we will discuss some evaluations for improper integrals using Theorem 1 for spe-
cific values of the parameters appearing in equations (2.1)-(2.3). As by-products, in the following
examples, interesting integrals are obtained.

We should point out that we consider the case |λ | = 1 in the next illustrative examples. In this
way, from equation (2.4), we can choose σ = 2 > σ0 = 1

1
α = 1. Equation (2.5) with σ = 2 imply

that

r cosθ = 2 and r sinθ = τ. (3.1)

By equation (3.1), we have

r =
√

4+ τ2 and θ = arccos
(

2√
4+ τ2

)
= arcsin

(
τ√

4+ τ2

)
. (3.2)

Example 1. We consider the function:

Eγ

1,β (−t) =
1

Γ(β )
1F1 (γ;β ;−t) ,

where 1F1 (γ;β ; t) be a confluent hypergeometric function [18]. In particular, if 0 < γ ≤ β ≤ 1,
the function ϕ(t) = tβ−1Eγ

1,β (−t) is CM.

From equation (2.1), we can derive that

Eγ

1,β (−t) =
2 t1−β e2 t

π

∫
∞

0

rγ−β

r̃
cos
[
θ (γ−β )− θ̃

]
cos(tτ)dτ, (3.3)

or in a different form, we can obtain an integral representation for confluent hypergeometric
function as follows:

1F1 (γ;β ;−t) =
2Γ(β ) t1−β e2 t

π

∫
∞

0

rγ−β

r̃
cos
[
θ (γ−β )− θ̃

]
cos(tτ)dτ, (3.4)

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GRIGOLETTO, OLIVEIRA and CAMARGO 223

where, according to equation (2.6),

θ̃ = γ arctan
(

τ

4

)
and r̃ =

(
4+ τ

2)γ/2
. (3.5)

Substituting equations (3.2) and (3.5) into equation (3.4), we obtain the result

1F1 (γ;β ;−t) =
2Γ(β ) t1−β e2 t

π

∫
∞

0
φ (t,τ) cos(tτ) dτ, (3.6)

for t > 0, where

φ (t,τ) =
1(

4+ τ2
)β/2

cos
[
(γ−β )arccos

(
2√

4+ τ2

)
− γ arctan

(
τ

4

)]
. (3.7)

Taking γ = β in equation (3.7), we have

1F1 (β ;β ;−t) =
2Γ(β ) t1−β e2 t

π

∫
∞

0

cos
[
β arctan

(
τ

4

)]
cos(tτ)

(4+ τ2)β/2 dτ. (3.8)

According to equation (3.8) with Eβ

1,β (−t) =
1

Γ(β )
1F1 (β ;β ;−t) we thus have

Eβ

1,β (−t) =
2 t1−β e2t

π

∫
∞

0

cos
[
β arctan

(
τ

4

)]
cos(tτ)

(4+ τ2)
β

2

dτ, (3.9)

for t > 0.

If γ = 1 and β = 2, by equations (3.3) and (1.8),

E1,2(−t) =
1−e−t

t
=

2e2 t

t π

∫
∞

0

cos(θ + θ̃) cos(tτ)
r r̃

dτ, (3.10)

that is,
π

2
e−2t (1−e−t)= ∫ ∞

0

1
r2 r̃2

(
r cos θ r̃ cos θ̃ − r sin θ r̃ sin θ̃

)
cos(tτ) dτ. (3.11)

Equations (2.6) and (3.1) provided that

r̃ cos θ̃ = 3 and r̃ sin θ̃ = τ. (3.12)

Taking into account the equations (3.12) and (3.1), we can rewrite equation (3.11) in the
respective form:

π

2
e−2t (1−e−t)= ∫ ∞

0

(
6− τ2

)
cos(tτ)

τ4 +13τ2 +36
dτ, for t > 0. (3.13)

Example 2. In this example we consider the function: Eα,α (−tα). In particular, if 0 < α ≤ 1, the
function ϕ(t) = tα−1Eα,α (−tα) is CM.

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224 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

In this case, if we use equation (2.3), since αγ−β = 0, then we have

Eα,α(−tα) =
2 t1−α e2t

π

∫
∞

0

sin θ̃ sin(tτ)
r̃

dτ, (3.14)

where t > 0 and θ̃ and r̃ are defined in equation (2.6), given by

r̃ =
√

r2α +2rα cos(θα)+1 and sin θ̃ =
rα sin(θα)

r̃
, (3.15)

where r and θ are given by equation (3.1).

When α = 2, then equation (2.6), in accordance with equation (3.1), yields the following formula

r̃ cos θ̃ = r2 cos(2θ)+1 = 5− τ
2 and r̃ sin θ̃ = r2 sin(2θ) = 4τ. (3.16)

Multiplying the integrand in equation (3.14) by
r̃
r̃

, and substituting equations (3.1) and (3.16)
into (3.14), we thus derive the following integral representation

E2,2(−t2) =
2e2t

t π

∫
∞

0

4τ sin(tτ)
τ4 +6τ2 +25

dτ, for t > 0. (3.17)

Equation (1.7) imply the following result

E2,2
(
−t2)= sinh it

it
=

sin t
t

.

Then, equation (3.17) can be rewritten in the alternative form:

e−2t sin t =
2
π

∫
∞

0

4τ sin(tτ)
τ4 +6τ2 +25

dτ, for t > 0. (3.18)

Example 3. In the last case we consider the function: Eα (−tα). In particular, if 0 < α ≤ 1,
the function ϕ(t) = Eα (−tα) is CM. From equation (2.2), we obtain the following integral
representation

Eα(−tα) =
2e2t

π

∫
∞

0

rα−1

r̃
cos
[
θ (α−1)− θ̃

]
cos(tτ)dτ, (3.19)

for t > 0, where r and θ are given by equation (3.1) and r̃ and θ̃ are given by equation (2.6).

Moreover, when manipulating the mathematical expression in equation (3.19), we can give
another similar integral representation as follows

Eα(−tα) =
2e2t

π

∫
∞

0

rα−2 [2rα +2 cos(θα)+ τ sin(θα)] cos(tτ)
r2α +2rα cos(θα)+1

dτ. (3.20)

In particular, when α = 1 in equation (3.20), we obtain another integral representation for the
exponential function:

π

2
e−t =

∫
∞

0

3 cos
( tτ

3

)
9+ τ2 dτ, for t > 0. (3.21)

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GRIGOLETTO, OLIVEIRA and CAMARGO 225

When α = 2, by using equation (1.9) and the integral representation type in equation (2.3), the
cosine function takes the form:

cos t = E2(−t2) =
2e2t

π

∫
∞

0

r
r̃

sin(θ̃ −θ) sin(tτ)dτ. (3.22)

The integral in equation (3.22) can be simplified to

cos t =
2e2t

π

∫
∞

0

(
τ3 +3τ

)
sin(tτ)

τ4 +6τ2 +25
dτ, for t > 0. (3.23)

Furthermore, using the relation in equation (1.10) and the equation (2.3), the hyperbolic cosine
function can be represented by

cosh t =
2e2t

π

∫
∞

0

(
τ3 +5τ

)
sin(tτ)

τ4 +10τ2 +9
dτ, for t > 0. (3.24)

Finally, we can use the relation in equation (1.6) and the above results to express the function

E4
(
t4)= 1

2
[
E2
(
t2)+E2

(
−t2)]= cos t + cosh t

2
.

In fact, equations (3.23) and (3.24) provided that

E4
(
t4)= e2t

π

∫
∞

0

[ (
τ3 +3τ

)
τ4 +6τ2 +25

+

(
τ3 +5τ

)
τ4 +10τ2 +9

]
sin(tτ)dτ, (3.25)

for t > 0.

4 CONCLUDING REMARKS

We build similar integral representations for the three-parameter Mittag-Leffler function on the
positive real axis using the method for finding inverse Laplace transform without using inte-
gration on the complex plane. Many authors have demonstrated interest in the study of the
asymptotic behavior of the Mittag-Leffler functions on the interpretation of the solutions of
problems associated with fractional diffusion. In this way the integral representations presented
in this paper can be used to analyze the asymptotic behavior of these functions. Furthermore,
this representation can express improper integrals in terms of trigonometric functions by means
of the Mittag-Leffler functions and the presented examples complement corresponding integral
representations.

ACKNOWLEDGMENT

The authors thank the referees for their valuable and constructive comments in relation to this
work and thank the collaboration of the members of our research group CF@FC.

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226 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

RESUMO. Através do método para encontrar a transformada de Laplace inversa sem o
uso de um contorno de integração no plano complexo, mostramos que a função de Mittag-
Leffler de três parâmetros, que aparece em muitos problemas associados com cálculo fra-
cionário, possui representações integrais similares no semieixo real positivo. Algumas delas
são apresentadas.

Palavras-chave: transformada de Laplace inversa, funções de Mittag-Leffler,
representações integrais, cálculo fracionário.

REFERENCES

[1] R.E. Bellman & R.S. Roth. “The Laplace Transform”. World Scientific, Singapore (1984).

[2] M.N. Berberan-Santos. Analytical inversion of the Laplace transform without contour integration:
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228 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION

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Tend. Mat. Apl. Comput., 20, N. 2 (2019)


	Preliminaries
	Mittag-Leffler functions
	Inversion of the Laplace transform

	Integral representations of Mittag-Leffler function
	Evaluation of a class of improper integrals
	Concluding remarks