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Table of Contents:

• Machine generated contents note: 2.1. Few Biographical Notes on Gosta Magnus Mittag-Leffier
• 2.2. Contents of the Five Papers by Mittag-Leffier on New Functions
• 2.3. Further History of Mittag-Leffler Functions
• 3.1. Definition and Basic Properties
• 3.2. Relations to Elementary and Special Functions
• 3.3. Recurrence and Differential Relations
• 3.4. Integral Representations and Asymptotics
• 3.5. Distribution of Zeros
• 3.6. Further Analytic Properties
• 3.7. Mittag-Leffier Function of a Real Variable
• 3.7.1. Integral Transforms
• 3.7.2. Complete Monotonicity Property
• 3.7.3. Relation to Fractional Calculus
• 3.8. Historical and Bibliographical Notes
• 3.9. Exercises
• 4.1. Series Representation and Properties of Coefficients
• 4.2. Explicit Formulas: Relations to Elementary and Special Functions
• 4.3. Differential and Recurrence Relations
• 4.4. Integral Relations and Asymptotics
• 4.5. Two-Parametric Mittag-Leffler Function as an Entire Function

• 4.6. Distribution of Zeros
• 4.7. Computations with the Two-Parametric Mittag-Leffler Function
• 4.8. Extension for Negative Values of the First Parameter
• 4.9. Further Analytic Properties
• 4.10. Two-Parametric Mittag-Leffler Function of a Real Variable
• 4.10.1. Integral Transforms of the Two-Parametric Mittag-Leffler Function
• 4.10.2. Complete Monotonicity Property
• 4.10.3. Relations to the Fractional Calculus
• 4.11. Historical and Bibliographical Notes
• 4.12. Exercises
• 5.1. Prabhakar (Three-Parametric Mittag-Leffler) Function
• 5.1.1. Definition and Basic Properties
• 5.1.2. Integral Representations and Asymptotics
• 5.1.3. Integral Transforms of the Prabhakar Function
• 5.1.4. Fractional Integrals and Derivatives of the Prabhakar Function
• 5.1.5. Relations to the Wright Function, H-Function and Other Special Functions
• 5.2. Generalized (Kilbas[—]Saigo) Mittag-Leffler Type Functions
• 5.2.1. Definition and Basic Properties
• 5.2.2. Order and Type of the Entire Function Eα,m,l(z)
• 5.2.3. Recurrence Relations for Eα,m,l(z)
• 5.2.4. Connection of En,m,l(z) with Functions of Hypergeometric Type
• 5.2.5. Differentiation Properties of En,m,l(z)
• 5.2.6. Fractional Integration of the Generalized Mittag-Leffler Function
• 5.2.7. Fractional Differentiation of the Generalized Mittag-Leffler Function
• 5.3. Historical and Bibliographical Notes
• 5.4. Exercises
• 6.1. Four-Parametric Mittag-Leffler Function: Luchko[—]Kilbas[—]Kiryakova's Approach
• 6.1.1. Definition and Special Cases
• 6.1.2. Basic Properties
• 6.1.3. Integral Representations and Asymptotics
• 6.1.4. Extended Four-Parametric Mittag-Leffler Functions
• 6.1.5. Relations to the Wright Function and the H-Function
• 6.1.6. Integral Transforms of the Four-Parametric Mittag-Leffler Function
• 6.1.7. Integral Transforms with the Four-Parametric Mittag-Leffler Function in the Kernel
• 6.1.8. Relations to the Fractional Calculus
• 6.2. Mittag-Leffler Functions with 2n Parameters
• 6.2.1. Definition and Basic Properties
• 6.2.2. Representations in Terms of Hypergeometric Functions
• 6.2.3. Integral Representations and Asymptotics
• 6.2.4. Extension of the 2n-Parametric Mittag-Leffler Function
• 6.2.5. Relations to the Wright Function and to the H-Function
• 6.2.6. Integral Transforms with the Multi-parametric Mittag-Leffler Functions
• 6.2.7. Relations to the Fractional Calculus
• 6.3. Historical and Bibliographical Notes
• 6.4. Exercises
• 7.1. Fractional Order Integral Equations
• 7.1.1. Abel Integral Equation
• 7.1.2. Other Integral Equations Whose Solutions Are Represented via Generalized Mittag-Leffler Functions
• 7.2. Fractional Ordinary Differential Equations
• 7.2.1. Fractional Ordinary Differential Equations with Constant Coefficients
• 7.2.2. Ordinary FDEs with Variable Coefficients
• 7.2.3. Other Types of Ordinary Fractional Differential Equations
• 7.3. Differential Equations with Fractional Partial Derivatives
• 7.3.1. Cauchy-Type Problems for Differential Equations with Fractional Partial Derivatives
• 7.3.2. Cauchy Problem for Differential Equations with Fractional Partial Derivatives
• 7.4. Historical and Bibliographical Notes
• 7.5. Exercises
• 8.1. Fractional Relaxation and Oscillations
• 8.1.1. Simple Fractional Relaxation and Oscillation
• 8.1.2. Composite Fractional Relaxation and Oscillations
• 8.2. Examples of Applications of the Fractional Calculus in Physical Models
• 8.2.1. Linear Visco-Elasticity
• 8.2.2. Other Deterministic Fractional Models
• 8.3. Historical and Bibliographical Notes
• 8.3.1. General Notes
• 8.3.2. Notes on Fractional Differential Equations
• 8.3.3. Notes on the Fractional Calculus in Linear Viscoelasticity
• 8.4. Exercises
• 9.1. Introduction
• 9.2. Mittag-Leffler Process According to Pillai
• 9.3. Elements of Renewal Theory and Continuous Time Random Walk (CTRW)
• 9.3.1. Renewal Processes
• 9.3.2. Continuous Time Random Walk (CTRW)
• 9.3.3. Renewal Process as a Special CTRW
• 9.4. Poisson Process and Its Fractional Generalization (the Renewal Process of Mittag-Leffler Type)
• 9.4.1. Mittag-Leffler Waiting Time Density
• 9.4.2. Poisson Process
• 9.4.3. Renewal Process of Mittag-Leffler Type
• 9.4.4. Thinning of a Renewal Process
• 9.5. Fractional Diffusion Process
• 9.5.1. Renewal Process with Reward
• 9.5.2. Limit of the Mittag-Leffler Renewal Process
• 9.5.3. Subordination in the Space-Time Fractional Diffusion Equation
• 9.5.4. Rescaling and Respeeding Concept Revisited: Universality of the Mittag-Leffler Density
• 9.6. Historical and Bibliographical Notes
• 9.7. Exercises
• A.1. Gamma Function
• A.1.1. Analytic Continuation
• A.1.2. Graph of the Gamma Function on the Real Axis
• A.1.3. Reflection or Complementary Formula
• A.1.4. Multiplication Formulas
• A.1.5. Pochhammer's Symbols
• A.1.6. Hankel Integral Representations
• A.1.7. Notable Integrals via the Gamma Function
• A.1.8. Asymptotic Formulas
• A.1.9. Infinite Products
• A.2. Beta Function
• A.2.1. Euler's Integral Representation
• A.2.2. Symmetry
• A.2.3. Trigonometric Integral Representation
• A.2.4. Relation to the Gamma Function
• A.2.5. Other Integral Representations
• A.2.6. Notable Integrals via the Beta Function
• A.3. Historical and Bibliographical Notes
• A.4. Exercises
• B.1. Definition and Series Representations
• B.2. Growth of Entire Functions: Order, Type and Indicator Function
• B.3. Weierstrass Canonical Representation: Distribution of Zeros
• B.4. Entire Functions of Completely Regular Growth
• B.5. Historical and Bibliographical Notes
• B.6. Exercises
• C.1. Fourier Type Transforms
• C.2. Laplace Transform
• C.3. Mellin Transform
• C.4. Simple Examples and Tables of Transforms of Basic Elementary and Special Functions
• C.5. Historical and Bibliographical Notes
• C.6. Exercises
• D.1. Definition: Contour of Integration
• D.2. Asymptotic Methods for the Mellin[—]Barnes Integral
• D.3. Historical and Bibliographical Notes
• D.4. Exercises
• E.1. Introduction to the Riemann[—]Liouville Fractional Calculus
• E.2. Liouville[—]Weyl Fractional Calculus
• E.3. Abel[—]Riemann Fractional Calculus
• E.3.1. Abel[—]Riemann Fractional Integrals and Derivatives
• E.4. Caputo Fractional Calculus
• E.4.1. Caputo Fractional Derivative
• E.5. Riesz[—]Feller Fractional Calculus
• E.5.1. Riesz Fractional Integrals and Derivatives
• E.5.2. Feller Fractional Integrals and Derivatives
• E.6. Grünwald[—]Letnikov Fractional Calculus
• E.6.1. Grünwald[—]Letnikov
• Approximation in the Riemann[—]Liouville Fractional Calculus
• E.6.2. Grünwald[—]Letnikov Approximation in the Riesz[—]Feller Fractional Calculus
• E.7. Historical and Bibliographical Notes
• F.1. Hypergeometric Functions
• F.1.1. Classical Gauss Hypergeometric Functions
• F.1.2. Euler Integral Representation: Mellin[—]Barnes Integral Representation
• F.1.3. Basic Properties of Hypergeometric Functions
• F.1.4. Hypergeometric Differential Equation
• F.1.5. Kummer's and Tricomi's Confluent Hypergeometric Functions
• F.1.6. Generalized Hypergeometric Functions and Their Properties
• F.2. Wright Functions
• F.2.1. Classical Wright Function
• F.2.2. Mellin[—]Barnes Integral Representation and Asymptotics
• F.2.3. Bessel[—]Wright Function: Generalized Wright Functions and Fox[—]Wright Functions
• F.3. Meijer G-Functions
• F.3.1. Definition via Integrals: Existence
• F.3.2. Basic Properties of the Meijer G-Functions
• F.3.3. Special Cases
• F.3.4. Relations to Fractional Calculus
• F.3.5. Integral Transforms of G-Functions
• F.4. Fox H-Functions
• F.4.1. Definition via Integrals: Existence
• F.4.2. Series Representations and Asymptotics: Recurrence Relations
• F.4.3. Special Cases
• F.4.4. Relations to Fractional Calculus
• F.4.5. Integral Transforms of H-Functions
• F.5. Historical and Bibliographical Notes
• F.6. Exercises.

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Author/Creator:

Gorenflo, Rudolf , author

Languages:

English

Language Notes:

Item content: English

Main Work:

Mittag-Leffler functions, related topics and applications [by Gorenflo, R.]

Related Series:

Springer monographs in mathematics

Subjects:

Functional equations

Functional analysis

General Notes:

Includes bibliographical references and index.
Description based on: Online resource; title from PDF title page (SpringerLink, viewed November 6, 2014).

Physical Description:

1 online resource.

Call Numbers:

QA431 .G67 2014eb

ISBNs:

9783662439302 (electronic bk.)
3662439301 (electronic bk.)
9783662439296 [Invalid]

OCLC Numbers:

894508489

Other Control Numbers:

EBC1967828 (source: MiAaPQ)
[Unknown Type]: ybp12143790