Mittag-Leffler Functions, Related Topics and Applications
Theory and Applications
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Chapter
Introduction
This book is devoted to an extended description of the properties of the Mittag-Leffler function, its numerous generalizations and their applications in different areas of modern science.
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Chapter
The Classical Mittag-Leffler Function
In this chapter we present the basic properties of the classical Mittag-Leffler function \(E_\alpha (z)\) E α ( z ) (see (1.0.1)). The material can be formally divided into two parts. Starting from t...
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Chapter
Mittag-Leffler Functions with Three Parameters
The Prabhakar generalized Mittag-Leffler function [Pra71] is defined where \((\gamma )_n = \gamma (\gamma +1)\ldots (\gamma +n-1)\) ( γ ) n = γ ( γ + 1 ) … ( γ + n - 1 ) (see formula (A.17) in App...
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Chapter
The Classical Wright Function
This chapter deals with the classical Wright function. Like the functions of Mittag-Leffler type, the functions of Wright type are known to play fundamental roles in various applications of the fractional calc...
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Chapter
Applications to Deterministic Models
Here we present material illuminating the role of the Mittag-Leffler function and its generalizations in the study of deterministic models. It has already been mentioned that the Mittag-Leffler function is clo...
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Chapter
Historical Overview of the Mittag-Leffler Functions
Gösta Magnus Mittag- was born on March 16, 1846, in Stockholm, Sweden. His father, John Olof Leffler, was a school teacher, and was also elected as a member of the Swedish Parliament.
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Chapter
Applications to Stochastic Models
This chapter is devoted to the application of the Mittag-Leffler function and related special functions in the study of certain stochastic processes. As this topic is so wide, we restrict our attention to some...
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Chapter
The Two-Parametric Mittag-Leffler Function
In this chapter we present the basic properties of the two-parametric Mittag-Leffler function \(E_{\alpha ,\beta } (z)\) E α , β ( z ) (see (1.0.3)), which is the most straightforward generalizatio...
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Chapter
Multi-index and Multi-variable Mittag-Leffler Functions
Consider the function defined for \(\alpha _1,\ \alpha _2\in {\mathbb R}\) α 1 , α 2 ∈ R \((\alpha _1^2+\alpha _2^2\ne 0)\) ( α 1 2 + α 2 2 ≠ 0 ) and \(\beta _1,\beta _2 \in {\mathbb C}\)
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Chapter
Applications to Fractional Order Equations
In this chapter we consider a number of integral equations and differential equations (mainly of fractional order). In representations of their solution, the Mittag-Leffler function, its generalizations and so...
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Book
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Chapter
Historical Overview of the Mittag-Leffler Functions
Gösta Magnus Mittag-Leffler was born on March 16, 1846, in Stockholm, Sweden. His father, John Olof Leffler, was a school teacher, and was also elected as a member of the Swedish Parliament. His mother, Gust...
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Chapter
The Two-Parametric Mittag-Leffler Function
In this chapter we present the basic properties of the two-parametric Mittag-Leffler function E α, β (z) (see (1.0.3
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Chapter
Multi-index Mittag-Leffler Functions
Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α 1 2 +α 2 2 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series
6.1.1 $$\displaystyle{ E_{\alpha _{... -
Chapter
Applications to Deterministic Models
Here we present material illuminating the role of the Mittag-Leffler function and its generalizations in the study of deterministic models. It has already been mentioned that the Mittag-Leffler function is clo...
-
Chapter
Introduction
The book is devoted to an extended description of the properties of the Mittag-Leffler function, its numerous generalizations and their applications in different areas of modern science.
-
Chapter
The Classical Mittag-Leffler Function
In this chapter we present the basic properties of the classical Mittag-Leffler function E α (z) (see (1.0.1)). The material can be formally divided into two parts.
-
Chapter
Mittag-Leffler Functions with Three Parameters
The Prabhakar generalized Mittag-Leffler function [Pra71] is defined as
5.1.1 $$\displaystyle{ E_{\alpha,\beta }^{\gamma }(z):=\sum _{ n=0}^{\infty } \frac{(\gamma )_{n}} {n!\varGamma (\alpha n+\beta )}\,z^... -
Chapter
Applications to Fractional Order Equations
In this chapter we consider a number of integral equations and differential equations (mainly of fractional order). In representations of their solution, the Mittag-Leffler function, its generalizations and so...
-
Chapter
Applications to Stochastic Models
This chapter is devoted to the application of the Mittag-Leffler function and related special functions in the study of certain stochastic processes. As this topic is so wide, we restrict our attention to some...
