Mittag-Leffler function

Last updated
The Mittag-Leffler function can be used to interpolate continuously between a Gaussian and a Lorentzian function. MittagLeffler GaussToLorentz.gif
The Mittag-Leffler function can be used to interpolate continuously between a Gaussian and a Lorentzian function.

In mathematics, the Mittag-Leffler functions are a family of special functions. They are complex-valued functions of a complex argument z, and moreover depend on one or two complex parameters.

Contents

The one-parameter Mittag-Leffler function, introduced by Gösta Mittag-Leffler in 1903, [1] [2] can be defined by the Maclaurin series

where is the gamma function, and is a complex parameter with .

The two-parameter Mittag-Leffler function, introduced by Wiman in 1905, [3] [2] is occasionally called the generalized Mittag-Leffler function. It has an additional complex parameter , and may be defined by the series [2] [4]

When , the one-parameter function is recovered.

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 class of functions are important in the theory of the fractional calculus.

See below for three-parameter generalizations.

Some basic properties

For , the Mittag-Leffler function is an entire function of order , and type for any value of . In some sense, the Mittag-Leffler function is the simplest entire function of its order. The indicator function of is [5] :50 This result actually holds for as well with some restrictions on when . [6] :67

The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of [2] )

from which the following asymptotic expansion holds : for and real such that then for all , we can show the following asymptotic expansions (Section 6. of [2] ):

-as :

,

-and as :

.

A simpler estimate that can often be useful is given, thanks to the fact that the order and type of is and , respectively: [6] :62

for any positive and any .

Special cases

For , the series above equals the Taylor expansion of the geometric series and consequently .

For we find: (Section 2 of [2] )

Error function:

Exponential function:

Hyperbolic cosine:

For , we have

For , the integral

gives, respectively: , , .

Mittag-Leffler's integral representation

The integral representation of the Mittag-Leffler function is (Section 6 of [2] )

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 [2] with )

Three-parameter generalizations

One generalization, characterized by three parameters, is

where and are complex parameters and . [6]

Another generalization is the Prabhakar function

where is the Pochhammer symbol.

Applications of Mittag-Leffler function

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]

See also

Notes

Related Research Articles

<span class="mw-page-title-main">Bessel function</span> Families of solutions to related differential equations

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number , which represents the order of the Bessel function. Although and produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of .

<span class="mw-page-title-main">Beta function</span> Mathematical function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral

<span class="mw-page-title-main">Landau distribution</span>

In probability theory, the Landau distribution is a probability distribution named after Lev Landau. Because of the distribution's "fat" tail, the moments of the distribution, such as mean or variance, are undefined. The distribution is a particular case of stable distribution.

<span class="mw-page-title-main">Stable distribution</span> Distribution of variables which satisfies a stability property under linear combinations

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

<span class="mw-page-title-main">Inverse-gamma distribution</span> Two-parameter family of continuous probability distributions

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.

The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution.

A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.

<span class="mw-page-title-main">Half-normal distribution</span> Probability distribution

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

<span class="mw-page-title-main">Normal-inverse-gamma distribution</span>

In probability theory and statistics, the normal-inverse-gamma distribution is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product is a product distribution.

A geometric stable distribution or geo-stable distribution is a type of leptokurtic probability distribution. Geometric stable distributions were introduced in Klebanov, L. B., Maniya, G. M., and Melamed, I. A. (1985). A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables. These distributions are analogues for stable distributions for the case when the number of summands is random, independent of the distribution of summand, and having geometric distribution. The geometric stable distribution may be symmetric or asymmetric. A symmetric geometric stable distribution is also referred to as a Linnik distribution. The Laplace distribution and asymmetric Laplace distribution are special cases of the geometric stable distribution. The Mittag-Leffler distribution is also a special case of a geometric stable distribution.

In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935):

In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined in fractal geometry. Fractal derivatives were created for the study of anomalous diffusion, by which traditional approaches fail to factor in the fractal nature of the media. A fractal measure t is scaled according to tα. Such a derivative is local, in contrast to the similarly applied fractional derivative. Fractal calculus is formulated as a generalization of standard calculus.

The Mittag-Leffler distributions are two families of probability distributions on the half-line . They are parametrized by a real or . Both are defined with the Mittag-Leffler function, named after Gösta Mittag-Leffler.

In fluid dynamics, the Burgers vortex or Burgers–Rott vortex is an exact solution to the Navier–Stokes equations governing viscous flow, named after Jan Burgers and Nicholas Rott. The Burgers vortex describes a stationary, self-similar flow. An inward, radial flow, tends to concentrate vorticity in a narrow column around the symmetry axis, while an axial stretching causes the vorticity to increase. At the same time, viscous diffusion tends to spread the vorticity. The stationary Burgers vortex arises when the three effects are in balance.

<span class="mw-page-title-main">Stable count distribution</span> Probability distribution

In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn in his 2017 study of daily distributions of the S&P 500 and the VIX. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

In the field of mathematics known as complex analysis, the indicator function of an entire function indicates the rate of growth of the function in different directions.

Prabhakar function is a certain special function in mathematics introduced by the Indian mathematician Tilak Raj Prabhakar in a paper published in 1971. The function is a three-parameter generalization of the well known two-parameter Mittag-Leffler function in mathematics. The function was originally introduced to solve certain classes of integral equations. Later the function was found to have applications in the theory of fractional calculus and also in certain areas of physics.

References

  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.
  2. 1 2 3 4 5 6 7 8 Haubold,H J and Mathai,A M and Saxena,R K, J Appl Math 2011, 298628
  3. Anders Wiman, Über den Fundamentalsatz in der Teorie [sic] der Funktionen , Acta Math 29, 191-201 (1905).
  4. Weisstein, Eric W. "Mittag-Leffler Function". mathworld.wolfram.com. Retrieved 2019-09-11.
  5. Cartwright, M. L. (1962). Integral Functions. Cambridge Univ. Press. ISBN   052104586X.
  6. 1 2 3 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.
  7. Pritz, T. (2003). Five-parameter fractional derivative model for polymeric damping materials. Journal of Sound and Vibration, 265(5), 935-952.
  8. Nonnenmacher, T. F., & Glöckle, W. G. (1991). A fractional model for mechanical stress relaxation. Philosophical magazine letters, 64(2), 89-93.