Mittag-Leffler function

<h2 id="some-basic-properties">Some basic properties</h2>
For 
 
 
 
 α
 >
 0
 
 
 {\displaystyle \alpha >0}
 
, the Mittag-Leffler function 
 
 
 
 
 E
 
 α
 ,
 β
 
 
 (
 z
 )
 
 
 {\displaystyle E_{\alpha ,\beta }(z)}
 
 is an entire function of <a href="/facts/Entire_function/XNluKzu7">order</a> 
 
 
 
 1
 
 /
 
 α
 
 
 {\displaystyle 1/\alpha }
 
, and <a href="/facts/Entire_function/XNluKzu7">type</a> 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 for any value of 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
. In some sense, the Mittag-Leffler function is the simplest entire function of its order. The <a href="/facts/Indicator_function_(complex_analysis)/XWAkpyHu">indicator function</a> of 
 
 
 
 
 E
 
 α
 
 
 (
 z
 )
 
 
 {\displaystyle E_{\alpha }(z)}
 
 is<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>: 50

h
          
            
              E
              
                α
              
            
          
        
        (
        θ
        )
        =
        
          
            {
            
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        θ
                        α
                      
                    
                    )
                  
                  ,
                
                
                  
                    for 
                  
                  
                    |
                  
                  θ
                  
                    |
                  
                  ≤
                  
                    
                      1
                      2
                    
                  
                  α
                  π
                  ;
                
              
              
                
                  0
                  ,
                
                
                  
                    otherwise
                  
                  .
                
              
            
            
          
        
      
    
    {\displaystyle h_{E_{\alpha }}(\theta )={\begin{cases}\cos \left({\frac {\theta }{\alpha }}\right),&{\text{for }}|\theta |\leq {\frac {1}{2}}\alpha \pi ;\\0,&{\text{otherwise}}.\end{cases}}}

This result actually holds for 
 
 
 
 β
 ≠
 1
 
 
 {\displaystyle \beta \neq 1}
 
 as well with some restrictions on 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 when 
 
 
 
 α
 =
 1
 
 
 {\displaystyle \alpha =1}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: 67 
The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of <a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>)

E
          
            α
            ,
            β
          
        
        (
        z
        )
        =
        
          
            1
            z
          
        
        
          E
          
            α
            ,
            β
            −
            α
          
        
        (
        z
        )
        −
        
          
            1
            
              z
              Γ
              (
              β
              −
              α
              )
            
          
        
        ,
      
    
    {\displaystyle E_{\alpha ,\beta }(z)={\frac {1}{z}}E_{\alpha ,\beta -\alpha }(z)-{\frac {1}{z\Gamma (\beta -\alpha )}},}

from which the following <a href="/facts/Asymptotic_expansion/4W79BNve">asymptotic expansion</a> holds : for 
 
 
 
 0
 <
 α
 <
 2
 
 
 {\displaystyle 0<\alpha <2}
 
 and 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 real such that

π
 α
 
 2
 
 
 <
 μ
 <
 min
 (
 π
 ,
 π
 α
 )
 
 
 {\displaystyle {\frac {\pi \alpha }{2}}<\mu <\min(\pi ,\pi \alpha )}
 
 then for all 
 
 
 
 N
 ∈
 
 
 N
 
 
 ∗
 
 
 ,
 N
 ≠
 1
 
 
 {\displaystyle N\in \mathbb {N} ^{*},N\neq 1}
 
, we can show the following asymptotic expansions (Section 6. of <a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>):
-as 
 
 
 
 
 
 |
 
 z
 
 |
 
 →
 +
 ∞
 ,
 
 |
 
 
 arg
 
 (
 z
 )
 
 |
 
 ≤
 μ
 
 
 {\displaystyle \,|z|\to +\infty ,|{\text{arg}}(z)|\leq \mu }
 
:

E
 
 α
 
 
 (
 z
 )
 =
 
 
 1
 α
 
 
 exp
 ⁡
 (
 
 z
 
 
 1
 α
 
 
 
 )
 −
 
 ∑
 
 k
 =
 1
 
 
 N
 
 
 
 
 1
 
 
 z
 
 k
 
 
 
 Γ
 (
 1
 −
 α
 k
 )
 
 
 
 +
 O
 
 (
 
 
 1
 
 z
 
 N
 +
 1
 
 
 
 
 )
 
 
 
 {\displaystyle E_{\alpha }(z)={\frac {1}{\alpha }}\exp(z^{\frac {1}{\alpha }})-\sum \limits _{k=1}^{N}{\frac {1}{z^{k}\,\Gamma (1-\alpha k)}}+O\left({\frac {1}{z^{N+1}}}\right)}
 
,
-and as 
 
 
 
 
 
 |
 
 z
 
 |
 
 →
 +
 ∞
 ,
 μ
 ≤
 
 |
 
 
 arg
 
 (
 z
 )
 
 |
 
 ≤
 π
 
 
 {\displaystyle \,|z|\to +\infty ,\mu \leq |{\text{arg}}(z)|\leq \pi }
 
:

E
 
 α
 
 
 (
 z
 )
 =
 −
 
 ∑
 
 k
 =
 1
 
 
 N
 
 
 
 
 1
 
 
 z
 
 k
 
 
 Γ
 (
 1
 −
 α
 k
 )
 
 
 
 +
 O
 
 (
 
 
 1
 
 z
 
 N
 +
 1
 
 
 
 
 )
 
 
 
 {\displaystyle E_{\alpha }(z)=-\sum \limits _{k=1}^{N}{\frac {1}{z^{k}\Gamma (1-\alpha k)}}+O\left({\frac {1}{z^{N+1}}}\right)}
 
.
A simpler estimate that can often be useful is given, thanks to the fact that the order and type of 
 
 
 
 
 E
 
 α
 ,
 β
 
 
 (
 z
 )
 
 
 {\displaystyle E_{\alpha ,\beta }(z)}
 
 is 
 
 
 
 1
 
 /
 
 α
 
 
 {\displaystyle 1/\alpha }
 
 and 
 
 
 
 1
 
 
 {\displaystyle 1}
 
, respectively:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>: 62

|
        
        
          E
          
            α
            ,
            β
          
        
        (
        z
        )
        
          |
        
        ≤
        C
        exp
        ⁡
        
          (
          
            σ
            
              |
            
            z
            
              
                |
              
              
                1
                
                  /
                
                α
              
            
          
          )
        
      
    
    {\displaystyle |E_{\alpha ,\beta }(z)|\leq C\exp \left(\sigma |z|^{1/\alpha }\right)}

for any positive 
 
 
 
 C
 
 
 {\displaystyle C}
 
 and any 
 
 
 
 σ
 >
 1
 
 
 {\displaystyle \sigma >1}
 
.

<h2 id="special-cases">Special cases</h2>
For 
 
 
 
 α
 =
 0
 
 
 {\displaystyle \alpha =0}
 
, the series above equals the Taylor expansion of the <a href="/facts/Geometric_series/Ne5TBH57">geometric series</a> and consequently 
 
 
 
 
 E
 
 0
 ,
 β
 
 
 (
 z
 )
 =
 
 
 1
 
 Γ
 (
 β
 )
 
 
 
 
 
 1
 
 1
 −
 z
 
 
 
 
 
 {\displaystyle E_{0,\beta }(z)={\frac {1}{\Gamma (\beta )}}{\frac {1}{1-z}}}
 
.
For 
 
 
 
 α
 =
 1
 
 /
 
 2
 ,
 1
 ,
 2
 
 
 {\displaystyle \alpha =1/2,1,2}
 
 we find: (Section 2 of <a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>)
<a href="/facts/Error_function/Ca2H7pGr">Error function</a>:

E
          
            
              1
              2
            
          
        
        (
        z
        )
        =
        exp
        ⁡
        (
        
          z
          
            2
          
        
        )
        erfc
        ⁡
        (
        −
        z
        )
        .
      
    
    {\displaystyle E_{\frac {1}{2}}(z)=\exp(z^{2})\operatorname {erfc} (-z).}

<a href="/facts/Exponential_function/ewlYxvR2">Exponential function</a>:

E
          
            1
          
        
        (
        z
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          
            
              z
              
                k
              
            
            
              Γ
              (
              k
              +
              1
              )
            
          
        
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          
            
              z
              
                k
              
            
            
              k
              !
            
          
        
        =
        exp
        ⁡
        (
        z
        )
        .
      
    
    {\displaystyle E_{1}(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (k+1)}}=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=\exp(z).}

<a href="/facts/Hyperbolic_cosine/Y4Cb5S35">Hyperbolic cosine</a>:

E
          
            2
          
        
        (
        z
        )
        =
        cosh
        ⁡
        (
        
          
            z
          
        
        )
        ,
        
           and 
        
        
          E
          
            2
          
        
        (
        −
        
          z
          
            2
          
        
        )
        =
        cos
        ⁡
        (
        z
        )
        .
      
    
    {\displaystyle E_{2}(z)=\cosh({\sqrt {z}}),{\text{ and }}E_{2}(-z^{2})=\cos(z).}

For 
 
 
 
 β
 =
 2
 
 
 {\displaystyle \beta =2}
 
, we have

E
          
            1
            ,
            2
          
        
        (
        z
        )
        =
        
          
            
              
                e
                
                  z
                
              
              −
              1
            
            z
          
        
        ,
      
    
    {\displaystyle E_{1,2}(z)={\frac {e^{z}-1}{z}},}

E
          
            2
            ,
            2
          
        
        (
        z
        )
        =
        
          
            
              sinh
              ⁡
              (
              
                
                  z
                
              
              )
            
            
              z
            
          
        
        .
      
    
    {\displaystyle E_{2,2}(z)={\frac {\sinh({\sqrt {z}})}{\sqrt {z}}}.}

For 
 
 
 
 α
 =
 0
 ,
 1
 ,
 2
 
 
 {\displaystyle \alpha =0,1,2}
 
, the integral

∫
          
            0
          
          
            z
          
        
        
          E
          
            α
          
        
        (
        −
        
          s
          
            2
          
        
        )
        
        
          
            d
          
        
        s
      
    
    {\displaystyle \int _{0}^{z}E_{\alpha }(-s^{2})\,{\mathrm {d} }s}

gives, respectively: 
 
 
 
 arctan
 ⁡
 (
 z
 )
 
 
 {\displaystyle \arctan(z)}
 
, 
 
 
 
 
 
 
 
 π
 
 2
 
 
 
 erf
 ⁡
 (
 z
 )
 
 
 {\displaystyle {\tfrac {\sqrt {\pi }}{2}}\operatorname {erf} (z)}
 
, 
 
 
 
 sin
 ⁡
 (
 z
 )
 
 
 {\displaystyle \sin(z)}
 
.

<h2 id="mittag-lefflers-integral-representation">Mittag-Leffler's integral representation</h2>
The integral representation of the Mittag-Leffler function is (Section 6 of <a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>)

E
          
            α
            ,
            β
          
        
        (
        z
        )
        =
        
          
            1
            
              2
              π
              i
            
          
        
        
          ∮
          
            C
          
        
        
          
            
              
                t
                
                  α
                  −
                  β
                
              
              
                e
                
                  t
                
              
            
            
              
                t
                
                  α
                
              
              −
              z
            
          
        
        
        d
        t
        ,
        ℜ
        (
        α
        )
        >
        0
        ,
        ℜ
        (
        β
        )
        >
        0
        ,
      
    
    {\displaystyle E_{\alpha ,\beta }(z)={\frac {1}{2\pi i}}\oint _{C}{\frac {t^{\alpha -\beta }e^{t}}{t^{\alpha }-z}}\,dt,\Re (\alpha )>0,\Re (\beta )>0,}

where the contour 
 
 
 
 C
 
 
 {\displaystyle C}
 
 starts and ends at 
 
 
 
 −
 ∞
 
 
 {\displaystyle -\infty }
 
 and circles around the singularities and branch points of the integrand.
Related to the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> and <a href="/facts/Mittag-Leffler_summation/Kes0lPUs">Mittag-Leffler summation</a> is the expression (Eq (7.5) of <a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> with 
 
 
 
 m
 =
 0
 
 
 {\displaystyle m=0}
 
)

∫
          
            0
          
          
            ∞
          
        
        
          e
          
            −
            t
            z
          
        
        
          t
          
            β
            −
            1
          
        
        
          E
          
            α
            ,
            β
          
        
        (
        ±
        r
        
        
          t
          
            α
          
        
        )
        
        d
        t
        =
        
          
            
              z
              
                α
                −
                β
              
            
            
              
                z
                
                  α
                
              
              ∓
              r
            
          
        
        ,
        ℜ
        (
        z
        )
        >
        0
        ,
        ℜ
        (
        α
        )
        >
        0
        ,
        ℜ
        (
        β
        )
        >
        0.
      
    
    {\displaystyle \int _{0}^{\infty }e^{-tz}t^{\beta -1}E_{\alpha ,\beta }(\pm r\,t^{\alpha })\,dt={\frac {z^{\alpha -\beta }}{z^{\alpha }\mp r}},\Re (z)>0,\Re (\alpha )>0,\Re (\beta )>0.}

<h2 id="three-parameter-generalizations">Three-parameter generalizations</h2>
One generalization, characterized by three parameters, is

 
 
 
 
 E
 
 α
 ,
 β
 
 
 γ
 
 
 (
 z
 )
 =
 
 (
 
 
 1
 
 Γ
 (
 γ
 )
 
 
 
 )
 
 
 ∑
 
 k
 =
 1
 
 
 ∞
 
 
 
 
 
 Γ
 (
 γ
 +
 k
 )
 
 z
 
 k
 
 
 
 
 k
 !
 Γ
 (
 α
 k
 +
 β
 )
 
 
 
 ,
 
 
 {\displaystyle E_{\alpha ,\beta }^{\gamma }(z)=\left({\frac {1}{\Gamma (\gamma )}}\right)\sum \limits _{k=1}^{\infty }{\frac {\Gamma (\gamma +k)z^{k}}{k!\Gamma (\alpha k+\beta )}},}

where 
 
 
 
 α
 ,
 β
 
 
 {\displaystyle \alpha ,\beta }
 
 and 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 are complex parameters and 
 
 
 
 ℜ
 (
 α
 )
 >
 0
 
 
 {\displaystyle \Re (\alpha )>0}
 
.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
Another generalization is the <a href="/facts/Prabhakar_function/mBBnzoIE">Prabhakar function</a>

 
 
 
 
 E
 
 α
 ,
 β
 
 
 γ
 
 
 (
 z
 )
 =
 
 ∑
 
 k
 =
 0
 
 
 ∞
 
 
 
 
 
 (
 γ
 
 )
 
 k
 
 
 
 z
 
 k
 
 
 
 
 k
 !
 Γ
 (
 α
 k
 +
 β
 )
 
 
 
 ,
 
 
 {\displaystyle E_{\alpha ,\beta }^{\gamma }(z)=\sum _{k=0}^{\infty }{\frac {(\gamma )_{k}z^{k}}{k!\Gamma (\alpha k+\beta )}},}

where 
 
 
 
 (
 γ
 
 )
 
 k
 
 
 
 
 {\displaystyle (\gamma )_{k}}
 
 is the <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbol</a>.

<h2 id="applications-of-mittag-leffler-function">Applications of Mittag-Leffler function</h2>
One of the applications of the Mittag-Leffler function is in modeling fractional order viscoelastic materials. Experimental investigations into the time-dependent relaxation behavior of viscoelastic materials are characterized by a very fast decrease of the stress at the beginning of the relaxation process and an extremely slow decay for large times. It can even take a long time before a constant asymptotic value is reached. Therefore, a lot of Maxwell elements are required to describe relaxation behavior with sufficient accuracy. This ends in a difficult optimization problem in order to identify a large number of material parameters. On the other hand, over the years, the concept of fractional derivatives has been introduced to the theory of <a href="/facts/Viscoelasticity/pDhHgx2T">viscoelasticity</a>. Among these models, the <a href="/facts/Standard_linear_solid_model/zPVQNDYE">fractional Zener</a> model was found to be very effective to predict the dynamic nature of rubber-like materials with only a small number of material parameters. The solution of the corresponding constitutive equation leads to a relaxation function of the Mittag-Leffler type. It is defined by the <a href="/facts/Power_series/vYh6sTps">power series</a> with negative arguments. This function represents all essential properties of the relaxation process under the influence of an arbitrary and continuous signal with a jump at the origin.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Mittag-Leffler_summation/Kes0lPUs">Mittag-Leffler summation</a></li>
<li><a href="/facts/Mittag-Leffler_distribution/Aq6Rb4tp">Mittag-Leffler distribution</a></li></ul>
<h2 id="notes">Notes</h2>
<ul><li><a href="/facts/R_(programming_language)/LSrkr8K8">R</a> Package <a href="https://CRAN.R-project.org/package=MittagLeffleR">'MittagLeffleR' </a> by Gurtek Gill, Peter Straka. Implements the Mittag-Leffler function, distribution, random variate generation, and estimation.</li></ul>

<ul><li><a href="https://www.springer.com/gp/book/9783662439296">Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer, New York, 2014)</a> 443 pages <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-662-43929-6</li>
<li>Igor Podlubny (1998). "chapter 1". Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-558840-2.</li>
<li>Kai Diethelm (2010). "chapter 4". The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics. Heidelberg and New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-14573-5.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=8738&objectType=FILE">Mittag-Leffler function: MATLAB code</a></li>
<li><a href="https://arxiv.org/abs/0707.2582">Mittag-Leffler and stable random numbers: Continuous-time random walks and stochastic solution of space-time fractional diffusion equations</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Mittag-Leffler, M.G.: Sur la nouvelle fonction E(x). C. R. Acad. Sci. Paris 137, 554–558 (1903), and several more papers in the following years. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Anders Wiman, Über den Fundamentalsatz in der Teorie [sic] der Funktionen 
 
 
 
 
 E
 
 a
 
 
 (
 x
 )
 
 
 {\displaystyle E_{a}(x)}
 
, Acta Math 29, 191-201 (1905). <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Weisstein, Eric W. "Mittag-Leffler Function". mathworld.wolfram.com. Retrieved 2019-09-11. <a href="http://mathworld.wolfram.com/Mittag-LefflerFunction.html" target="_blank">http://mathworld.wolfram.com/Mittag-LefflerFunction.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Cartwright, M. L. (1962). Integral Functions. Cambridge Univ. Press. ISBN 052104586X. {{cite book}}: ISBN / Date incompatibility (help) <a href="052104586X" target="_blank">052104586X</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Gorenflo, Rudolf; Kilbas, Anatoly A.; Mainardi, Francesco; Rogosin, Sergei V. (2014). Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications. Springer Monographs in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-43930-2. ISBN 978-3-662-43929-6. <a href="978-3-662-43929-6" target="_blank">978-3-662-43929-6</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Gorenflo, Rudolf; Kilbas, Anatoly A.; Mainardi, Francesco; Rogosin, Sergei V. (2014). Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications. Springer Monographs in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-43930-2. ISBN 978-3-662-43929-6. <a href="978-3-662-43929-6" target="_blank">978-3-662-43929-6</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628 <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628" target="_blank">https://onlinelibrary.wiley.com/doi/epdf/10.1155/2011/298628</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Gorenflo, Rudolf; Kilbas, Anatoly A.; Mainardi, Francesco; Rogosin, Sergei V. (2014). Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications. Springer Monographs in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-43930-2. ISBN 978-3-662-43929-6. <a href="978-3-662-43929-6" target="_blank">978-3-662-43929-6</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Pritz, T. (2003). Five-parameter fractional derivative model for polymeric damping materials. Journal of Sound and Vibration, 265(5), 935-952. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Nonnenmacher, T. F., & Glöckle, W. G. (1991). A fractional model for mechanical stress relaxation. Philosophical magazine letters, 64(2), 89-93. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Mittag-Leffler function open-in-new

Mittag-Leffler function