This book provides the knowledge of the newly-established supertrigonometric and superhyperbolic functions with the special functions such as Mittag-Leffler, Wiman, Prabhakar, Miller-Ross, Rabotnov, Lorenzo-Hartley, Sonine, Wright and Kohlrausch-Williams-Watts functions, Gauss hypergeometric series and Clausen hypergeometric series. The special functions can be considered to represent a great many of the real-world phenomena in mathematical physics, engineering and other applied sciences. The audience benefits of new and original information and references in the areas of the special functions applied to model the complex problems with the power-law behaviors.
The results are important and interesting for scientists and engineers to represent the complex phenomena arising in applied sciences therefore graduate students and researchers in mathematics, physics and engineering might find this book appealing.
Author(s): Xiao-Jun Yang
Publisher: Springer
Year: 2022
Language: English
Pages: 916
City: Singapore
Preface
Acknowledgments
Contents
About the Author
1 Preliminaries
1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function, and Related Functions
1.1.1 The Euler Gamma Function
1.1.2 The Pochhammer Symbols and Related Formulas
1.1.3 The Euler Beta Function
1.1.4 The Extended Euler Gamma Functions
1.1.5 The Extended Euler Beta Functions
1.2 Gauss Hypergeometric Series and Supertrigonometric and Superhyperbolic Functions
1.2.1 The Gauss Hypergeometric Series
1.2.2 The Hypergeometric Supertrigonometric Functions via Gauss Superhyperbolic Series
1.2.3 The Hypergeometric Superhyperbolic Functions via Gauss Hypergeometric Series
1.3 Clausen Hypergeometric Series and Supertrigonometric and Superhyperbolic Functions
1.3.1 The Clausen Hypergeometric Series
1.3.2 The Hypergeometric Supertrigonometric Functions via Clausen Superhyperbolic Series
1.3.3 The Hypergeometric Superhyperbolic Functions via Clausen Superhyperbolic Series
1.3.4 The Series Representations for the Special Functions
1.4 The Laplace and Mellin Transforms
1.4.1 The Laplace Transforms for the Special Functions
1.4.2 The Mellin Transforms for the Special Functions
1.5 Calculus with Respect to Monotone Functions
1.5.1 The Newton–Leibniz Calculus
1.5.2 Calculus with Respect to Monotone Function
1.5.2.1 The Leibniz Derivative
1.5.2.2 The Riemann–Stieltjes Integrals
1.5.3 The Special Integral Equations
1.5.4 Generalized Functions and Anomalous Linear Viscoelasticity via Derivative with Respect to Another Function
1.5.4.1 Dirac Delta Function
1.5.4.2 Dirac-Like Delta Function of First Type
1.5.4.3 The Anomalous Linear Viscoelasticity via Derivative with Respect to Another Function
1.6 Derivative and Integral with Respect to Power-Law Function
1.6.1 The Derivative with Respect to Power-Law Function
1.6.2 The Integral with Respect to Power-Law Function
1.6.2.1 The Integral with Respect to Power-Law Function
1.6.2.2 The Dirac-Like Delta Function of Second Type
1.6.3 The Scaling-Law Calculus
1.6.3.1 The Scaling-Law Derivatives
1.6.3.2 The Scaling-Law Integrals
1.6.3.3 The Improper Scaling-Law Integral and Scaling-Law Integral Equations
1.6.3.4 The Anomalous Relaxation
1.6.3.5 The Anomalous Linear Viscoelasticity
1.6.3.6 The Dirac-Like Delta Function of Third Type
1.6.4 The Special Formulas via Scaling-Law Calculus
1.6.5 Other Calculus Operators with Respect to Monotone Functions
1.6.5.1 The Calculus with Respect to Exponential Function
1.6.5.2 The Calculus with Respect to Logarithmic Function
1.6.5.3 The Calculus with Respect to Complex Topology
2 Wright Function and Integral Transforms via Dunkl Transform
2.1 The Special Functions Related to Wright Function and Integral Representations
2.1.1 The Wright's Generalized Hypergeometric Function
2.1.2 The Integral Representations via Wright's Generalized Hypergeometric Function
2.1.3 The Integral Transforms for the Generalized Wright Functions
2.1.4 The Supertrigonometric Functions via Wright's Generalized Hypergeometric Function
2.1.5 The Superhyperbolic Functions via Wright's Generalized Hypergeometric Function
2.1.6 The Supertrigonometric Functions via Wright Function
2.1.7 The Superhyperbolic Functions via Wright Function
2.2 The Truncated Wright's Generalized Hypergeometric Function
2.3 The Integral Transforms via Dunkl Transform
2.3.1 The Dunkl Transform
2.3.2 New Integral Transforms of First Type
2.3.3 New Integral Transforms of Second Type
2.3.4 New Integral Transforms of Third Type
2.3.4.1 New Integral Transforms of First Type Containing the Wright Function
2.3.4.2 New Integral Transforms of Second Type Containing Wright Function
3 Mittag-Leffler, Supertrigonometric, and Superhyperbolic Functions
3.1 The Mittag-Leffler Function: History, Definitions, Properties, and Theorems
3.1.1 The Mittag-Leffler Function
3.1.2 Special Integral Representations
3.1.3 The Integral Transforms for the Mittag-Leffler Functions
3.1.4 The Supertrigonometric Functions via Mittag-Leffler Function
3.1.5 The Superhyperbolic Functions via Mittag-Leffler Function
3.1.6 The Pre-supertrigonometric Functions via Mittag-Leffler Function
3.1.7 The Pre-superhyperbolic Functions via Mittag-Leffler Function
3.1.8 The Laplace Transforms of the Special Functions via Mittag-Leffler Function
3.2 Analytic Number Theory Involving the Mittag-Leffler Function
3.2.1 The Basic Formulas Involving the Mittag-Leffler Function
3.2.2 The Generalized Hyperbolic Function
3.3 The Special Integral Equations via Mittag-Leffler Function and Related Functions
3.3.1 The Integral Equations of Volterra Type
3.3.2 The Integral Equations of Fredholm Type
3.4 The Integral Representations for the Special Function via Mittag-Leffler Function
3.5 The Fractional Equations via Mittag-Leffler Function and Related Functions
3.6 General Fractional Calculus Operators with Mittag-Leffler Function
3.6.1 Hille–Tamarkin General Fractional Derivative
3.6.2 Hille–Tamarkin General Fractional Integrals
3.6.3 Liouville–Weyl–Hille–Tamarkin Type General Fractional Calculus
3.6.4 Hilfer–Hille–Tamarkin Type General Fractional Derivative with Nonsingular Kernel
3.6.5 Hille–Tamarkin General Fractional Derivative with Respect to Another Function
3.6.6 Hille–Tamarkin General Fractional Integrals with Respect to Another Function
3.6.7 Liouville–Weyl–Hille–Tamarkin Type General Fractional Calculus with Respect to Another Function
3.6.8 Hilfer–Hille–Tamarkin Type General Fractional Derivative with Respect to Another Function
3.7 The Integral Representations Related to Mittag-Leffler Function
3.8 The Relationship Between Mittag-Leffler Function and Wright's Generalized Hypergeometric Function
3.9 The Truncated Mittag-Leffler Functions and Related Functions
3.10 Applications in Anomalous Linear Viscoelasticity
4 Wiman, Supertrigonometric, and Superhyperbolic Functions
4.1 The Wiman Function: History, Definitions, Properties, and Theorems
4.1.1 The Wiman Function
4.1.2 The Supertrigonometric Functions via Wiman Function
4.1.3 The Superhyperbolic Functions via Wiman Function
4.1.4 The Pre-supertrigonometric Functions via Wiman Function
4.1.5 The Pre-superhyperbolic Functions via Wiman Function
4.1.6 Some Special Cases via Wiman Function
4.1.6.1 Case 1: The Supertrigonometric and Superhyperbolic Functions with the Power Law
4.1.6.2 Case 2: The Supertrigonometric and Superhyperbolic Functions with the Parameter
4.1.6.3 Case 3: The Supertrigonometric and Superhyperbolic Functions with the Power Law and the Parameter
4.1.7 The Special Integral Equations via Viman Function and Related Functions
4.1.7.1 Integral Equations of Volterra Type
4.1.7.2 Integral Equations of Fredholm Type
4.1.8 The Integral Representations Related to Viman Function
4.1.9 The Special Cases Based on the Wiman Function
4.2 The Integral Representations Related to Wiman, Supertrigonometric, and Superhyperbolic Functions
4.3 The Truncated Wiman Functions and Related Functions
4.4 General Fractional Derivatives Within the Wiman Kernel
4.4.1 General Fractional Derivatives Within the Wiman Kernel
4.4.2 Hilfer-Type General Fractional Derivatives with the Wiman Kernel
4.4.3 General Fractional Derivatives with Respect to Another Function via Wiman Function
4.5 Applications
5 Prabhakar, Supertrigonometric, and Superhyperbolic Functions
5.1 The Prabhakar Function: History, Definitions, Properties, and Theorems
5.1.1 The Prabhakar Function
5.1.2 The Supertrigonometric Functions via Prabhakar Function
5.1.3 The Superhyperbolic Functions via Prabhakar Function
5.1.4 The Pre-Supertrigonometric Functions via Prabhakar Function
5.1.5 The Pre-Superhyperbolic Functions via Prabhakar Function
5.1.6 The Pre-Supertrigonometric Functions with Power Law via Prabhakar Function
5.1.7 The Pre-Superhyperbolic Functions with Power Law via Prabhakar Function
5.1.8 The Pre-Supertrigonometric Functions with the Parameter via Prabhakar Function
5.1.9 The Pre-Superhyperbolic Functions with the Parameter via Prabhakar Function
5.1.10 The Pre-Supertrigonometric Functions with the Power Law and Parameter via Prabhakar Function
5.1.11 The Pre-Superhyperbolic Functions with the Power Law and Parameter via Prabhakar Function
5.2 The Integral Representations for Special Functions Related to Prabhakar Function
5.3 The Truncated Prabhakar Functions and Related Functions
5.3.1 The Truncated Prabhakar Functions
5.3.2 Other Special Functions Related to Prabhakar Function
5.4 General Fractional Calculus Operators via Prabhakar Function
5.4.1 Kilbas–Saigo–Saxena Derivative via Prabhakar Function
5.4.2 Garra–Gorenflo–Polito–Tomovski Derivative via Prabhakar Function
5.4.3 Prabhakar-Type Integrals
5.4.4 Kilbas–Saigo–Saxena-Type Derivative via Prabhakar Function
5.4.5 Garra–Gorenflo–Polito–Tomovski-Type Derivative via Prabhakar Function
5.4.6 Prabhakar-Type Integrals
5.4.7 Hilfer-Type Derivative via Prabhakar Function
5.4.8 Kilbas–Saigo–Saxena-Type Derivative with Respect to Another Function
5.4.9 Garra–Gorenflo–Polito–Tomovski-Type Derivative with Respect to Another Function
5.4.10 Prabhakar-Type Integrals with Respect to Another Function
5.4.11 Kilbas–Saigo–Saxena-Type Derivative with Respect to Another Function
5.4.12 Garra–Gorenflo–Polito–Tomovski-Type Derivative with Respect to Another Function
5.4.13 Prabhakar-Type Integrals with Respect to Another Function
5.4.14 Hilfer-Type Derivative with Respect to Another Function via Prabhakar Function
5.5 Applications
5.5.1 The Integral Equations in the Kernel of New Special Functions
5.5.2 Anomalous Viscoelasticity and Diffusion
6 Other Special Functions Related to Mittag-Leffler Function
6.1 The Sonine Functions: History, Definitions, and Properties
6.1.1 The Sonine Functions of First Type
6.1.2 The Supertrigonometric Functions via Sonine Function of First Type
6.1.3 The Superhyperbolic Functions via Sonine Function of First Type
6.1.4 The Integral Representations of the Supertrigonometric and Superhyperbolic Functions
6.1.5 The Sonine Functions of Second Type
6.1.6 The Supertrigonometric Functions via Sonine Function of Second Type
6.1.7 The Superhyperbolic Functions via Sonine Function of Second Type
6.1.8 The Sonine Function of Third Type
6.1.9 The Supertrigonometric Functions via Sonine Function of Third Type
6.1.10 The Superhyperbolic Functions via Sonine Function of Third Type
6.1.11 The Integral Representations Related to Sonine Function of Third Type
6.1.12 The Integral Representations for the Sonine Function of Second Type
6.2 The Rabotnov Fractional Exponential Function
6.2.1 The Rabotnov Fractional Exponential Function: History and Properties
6.2.2 The Supertrigonometric Functions via Rabotnov Function
6.2.3 The Superhyperbolic Functions via Rabotnov Function
6.2.4 The Supertrigonometric Functions via Rabotnov Type Function
6.2.5 The Superhyperbolic Functions via Rabotnov Type Function
6.2.6 The Integral Representations of the Supertrigonometric and Superhyperbolic Functions
6.2.7 The Gauss–Rabotnov Type Functions
6.3 The Miller–Ross Function
6.3.1 The Miller–Ross Function: History and Properties
6.3.2 The Supertrigonometric Functions via Miller–Ross Function
6.3.3 The Superhyperbolic Functions via Miller–Ross Function
6.3.4 The Integral Representations Related to Miller–Ross Function
6.4 The Lorenzo–Hartley Functions
6.4.1 The Lorenzo–Hartley Function of First Type
6.4.2 The Supertrigonometric Functions via Lorenzo–Hartley Function of First Type
6.4.3 The Superhyperbolic Functions via Lorenzo–Hartley Function of First Type
6.4.4 The Integral Representations for the Special Functions Related to the Lorenzo–Hartley Function of First Type
6.4.5 The Lorenzo–Hartley Function of Second Type
6.4.6 The Supertrigonometric Functions via Lorenzo–Hartley Function of Second Type
6.4.7 The Superhyperbolic Functions via Lorenzo–Hartley Function of Second Type
6.4.8 The Integral Representations for the Special Functions Related to the Lorenzo–Hartley Function of Second Type
7 Kohlrausch–Williams–Watts Function and Related Topics
7.1 The Kohlrausch–Williams–Watts Function: History, Definitions, and Properties
7.1.1 The Kohlrausch–Williams–Watts Function
7.1.2 The Subtrigonometric Functions via Kohlrausch–Williams–Watts Function
7.1.3 The Subhyperbolic Functions via Kohlrausch–Williams–Watts Function
7.1.4 The Subtrigonometric Functions via Kohlrausch–Williams–Watts Type Function
7.1.5 The Subhyperbolic Functions via Kohlrausch–Williams–Watts Type Function
7.1.6 The Integral Representations Associated with Kohlrausch–Williams–Watts Function
7.1.7 The Special Functions with Complex Topology
7.1.8 Subsurfaces and Geometric Representations Related to Kohlrausch–Williams–Watts Function
7.2 The Fourier-Type Series Theory via Subtrigonometric Series with Respect to Monotone Function
7.2.1 Theory of Fourier Series: History and Properties
7.2.2 The Subtrigonometric and Subhyperbolic Functions with Respect to Monotone Function
7.2.3 The Subtrigonometric Functions with Respect to Monotone Function
7.2.4 The Subhyperbolic Functions with Respect to Monotone Function
7.3 Theory of Subtrigonometric Series with Respect to Monotone Function
7.3.1 The Subtrigonometric Series with Respect to Monotone Function
7.3.2 The Subtrigonometric Series with Respect to Scaling-Law Function
7.3.3 Theory of Subtrigonometric Series with Respect to Complex and Power-Law Functions
7.3.4 Applications
7.4 The Fourier-Like Integral Transforms via Subtrigonometric Functions with Respect to Monotone Function
7.4.1 Fourier Transform: History, Concepts, and Theorems
7.4.2 The Integral Transform Operator with Respect to Monotone Function of First Type
7.4.3 The Integral Transform Operator with Respect to Monotone Function of Second Type
7.4.4 The Integral Transform Operator with Respect to Monotone Function of Third Type
7.4.5 The Integral Transform Operator with Respect to Monotone Function of Fourth Type
7.4.6 The Integral Transform Operator with Respect to Monotone Function of Fifth Type
7.4.7 The Integral Transform Operator with Respect to Monotone Function of Sixth Type
7.4.8 The Integral Transform Operator with Respect to Monotone Function of Seventh Type
7.4.9 The Integral Transform Operator with Respect to Monotone Function of Eighth Type
7.4.10 The Integral Transform Operator with Respect to Monotone Function of Ninth Type
7.4.11 The Integral Transform Operator with Respect to Power-Law Function of Second Type
7.4.12 The Integral Transform Operator with Respect to Power-Law Function of Fifth Type
7.4.13 The Integral Transform Operator with Respect to Scaling-Law Function of Second Type
7.4.14 The Integral Transform Operator with Respect to Scaling-Law Function of Fifth Type
7.4.15 Applications
7.5 The Laplace-Like Transforms via Subtrigonometric Functions with Respect to Monotone Function
7.5.1 Laplace Transform: History, Concepts, and Theorems
7.5.2 The Integral Transform Operator with Respect to Monotone Function of First Type
7.5.3 The Integral Transform Operator with Respect to Monotone Function of Second Type
7.5.4 The Integral Transform Operator with Respect to Monotone Function of Third Type
7.5.5 The Integral Transform Operator with Respect to Scaling-Law Function of Second Type
7.5.6 The Integral Transform Operator with Respect to Power-Law Function of Second Type
7.5.7 The Bilateral Integral Transform Operator with Respect to Monotone Function of Fourth Type
7.5.8 The Bilateral Integral Transform Operator with Respect to Monotone Function of Second Type
7.5.9 The Bilateral Integral Transform Operator with Respect to Monotone Function of Third Type
7.5.10 The Bilateral Integral Transform Operator with Respect to Scaling-Law Function of Second Type
7.5.11 The Bilateral Integral Transform Operator with Respect to Power-Law Function of Second Type
7.5.12 Applications
7.6 The Mellin-Like Transforms via Subtrigonometric Functions with Respect to Monotone Function
7.6.1 Mellin Transform: History, Concepts, and Theorems
7.6.2 The Integral Transform Operator with Respect to Monotone Function of First Type
7.6.3 The Integral Transform Operator with Respect to Monotone Function of Second Type
7.6.4 The Integral Transform Operator with Respect to Scaling-Law Function of Second Type
7.6.5 The Integral Transform Operator with Respect to Power-Law Function of Second Type
References
