MittagLeffler

Mittag-Leffler refers to a family of complex functions introduced by the Swedish mathematician Gösta Mittag-Leffler. The Mittag-Leffler function generalizes the exponential function and plays a central role in fractional calculus, complex analysis, and related areas of applied mathematics.

The two-parameter Mittag-Leffler function E_{α,β}(z) is defined by the convergent power series

E_{α,β}(z) = ∑_{k=0}^∞ z^k / Γ(α k + β),

where α > 0 and β are real (or complex) parameters. The one-parameter version is E_{α}(z) = E_{α,1}(z). A key special case is α = 1, for which E_{1}(z) = ∑_{k=0}^∞ z^k / k! = e^z. Thus the Mittag-Leffler function reduces to the exponential when α = 1.

Analytic properties and transforms: E_{α,β}(z) is an entire function of z for α > 0. A fundamental Laplace transform relation is

L{ t^{β-1} E_{α,β}(λ t^α) }(s) = s^{α-β} / (s^α − λ),

valid for suitable s with Re(s) > |λ|^{1/α}. This makes the function a natural generalization of the exponential kernel in linear differential equations of fractional order.

Applications: The Mittag-Leffler function appears prominently in solutions to linear fractional differential equations, modeling of anomalous diffusion and viscoelastic relaxation, and in certain stochastic processes (e.g., fractional Poisson processes). It provides a flexible analytical tool when standard exponentials are insufficient to capture memory and non-local behavior.

History: The function was introduced in the context of early 20th-century complex analysis by Gösta Mittag-Leffler and has since become a fundamental object in fractional calculus and related disciplines.