In mathematics, the Mittag-Leffler function is a special function, a complex function which depends on two complex parameters and . It may be defined by the following series when the real part of is strictly positive:[1][2]
where is the gamma function. When , it is abbreviated as . For , the series above equals the Taylor expansion of the geometric series and consequently .
In the case and are real and positive, the series converges for all values of the argument , so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus.
For , the Mittag-Leffler function is an entire function of order , and is in some sense the simplest entire function of its order.
The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of [1])
from which the following Poincaré asymptotic expansion holds : for and real such that then for all , we can show the following asymptotic expansions (Section 6. of [1]):
-as :
-and as :
where we used the notation .
The Mittag-Leffler function, characterized by three parameters, is expressed as follows:
where and are complex parameters and .[3]
For , the Mittag-Leffler function with three parameters is reformulated as:
where and it exhibits the following property:
.[4]
Additionally, a relation concerning the first parameter of the 2-parameter Mittag-Leffler function is as follows:
For we find: (Section 2 of [1])
The sum of a geometric progression:
For , we have
For , the integral
gives, respectively: , , .
The integral representation of the Mittag-Leffler function is (Section 6 of [1])
where the contour starts and ends at and circles around the singularities and branch points of the integrand.
Related to the Laplace transform and Mittag-Leffler summation is the expression (Eq (7.5) of [1] with )
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 viscoelasticity. Among these models, the fractional Zener 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 power series 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.[7][8]
This article incorporates material from Mittag-Leffler function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.