| Foreword | p. v |
| Acknowledgments | p. vii |
| Introduction | p. 1 |
| Functions | p. 5 |
| Gamma function | p. 6 |
| functions | p. 7 |
| Hypergeometric functions | p. 9 |
| Miscellaneous functions | p. 9 |
| The Fractional Derivative | p. 11 |
| Basics | p. 11 |
| The fractional Leibniz product rule | p. 16 |
| Discussion | p. 18 |
| Orthogonal polynomials | p. 18 |
| Differential representation of the Riemann fractional derivative | p. 20 |
| Friction Forces | p. 23 |
| Classical description | p. 23 |
| Fractional friction | p. 26 |
| Fractional Calculus | p. 33 |
| The Fourier transform | p. 34 |
| The fractional integral | p. 35 |
| The Liouville fractional integral | p. 36 |
| The Riemann fractional integral | p. 36 |
| Correlation of fractional integration and differentiation | p. 37 |
| The Liouville fractional derivative | p. 38 |
| The Riemann fractional derivative | p. 39 |
| The Liouville fractional derivative with inverted operator sequence: the Liouville-Caputo fractional derivative | p. 40 |
| The Riemann fractional derivative with inverted operator sequence: the Caputo fractional derivative | p. 42 |
| Fractional derivative of second order | p. 43 |
| The Riesz fractional derivative | p. 44 |
| The Feller fractional derivative | p. 46 |
| Fractional derivatives of higher orders | p. 47 |
| Geometric interpretation of the fractional integral | p. 51 |
| Low level fractionality | p. 53 |
| Discussion | p. 55 |
| Semigroup property of the fractional integral | p. 55 |
| The Fractional Harmonic Oscillator | p. 57 |
| The fractional harmonic oscillator | p. 58 |
| The harmonic oscillator according to Fourier | p. 58 |
| The harmonic oscillator according to Riemann | p. 60 |
| The harmonic oscillator according to Caputo | p. 62 |
| Wave Equations and Parity | p. 65 |
| Fractional wave equations | p. 65 |
| Parity and time-reversal | p. 67 |
| Solutions of the free regularized fractional wave equation | p. 68 |
| Nonlocality and Memory Effects | p. 75 |
| A short history of nonlocal concepts | p. 75 |
| From local to nonlocal operators | p. 77 |
| Memory effects | p. 88 |
| Quantum Mechanics | p. 93 |
| Canonical quantization | p. 95 |
| Quantization of the classical Hamilton function and free solutions | p. 95 |
| Temperature dependence of a fission yield and determination of the corresponding fission potential | p. 99 |
| The fractional Schrödinger equation with an infinite well potential | p. 102 |
| Radial solutions of the fractional Schrödinger equation | p. 107 |
| Fractional Spin: a Property of Particles Described with the Fractional Schrödinger Equation | p. 111 |
| Spin | p. 112 |
| Fractional spin | p. 113 |
| Factorization | p. 117 |
| The Dirac equation | p. 117 |
| The fractional Dirac equation | p. 118 |
| The fractional Pauli equation | p. 120 |
| Symmetries | p. 123 |
| Characteristics of fractional group theory | p. 124 |
| The fractional rotation group SON | p. 126 |
| The Fractional Symmetric Rigid Rotor | p. 133 |
| The spectrum of the fractional symmetric rigid rotor | p. 133 |
| Rotational limit | p. 136 |
| Vibrational limit | p. 137 |
| Davidson potential: the so called -unstable limit | p. 138 |
| Linear potential limit | p. 140 |
| The magic limit | p. 141 |
| Comparison with experimental data | p. 144 |
| q-deformed Lie Algebras and Fractional Calculus | p. 153 |
| q-deformed Lie algebras | p. 153 |
| The fractional q-deformed harmonic oscillator | p. 156 |
| The fractional q-deformed symmetric rotor | p. 160 |
| Half-integer representations of the fractional rotation group SO(3) | p. 162 |
| Fractional Spectroscopy of Hadrons | p. 165 |
| Phenomenology of the baryon spectrum | p. 166 |
| Charmonium | p. 171 |
| Phenomenology of meson spectra | p. 176 |
| Metaphysics: About the internal structure of quarks | p. 184 |
| Higher Dimensional Fractional Rotation Groups | p. 187 |
| The four decompositions of the mixed fractional SO (9) | p. 187 |
| Notation | p. 189 |
| The nine dimensional fractional Caputo-Riemann-Riemann symmetric rotor | p. 192 |
| Magic numbers of nuclei | p. 193 |
| Ground state properties of nuclei | p. 196 |
| Fine structure of the single particle spectrum: the extended Caputo-Riemann-Riemann symmetric rotor | p. 201 |
| Magic numbers of electronic clusters: the nine dimensional fractional Caputo-Caputo-Riemann symmetric rotor | p. 206 |
| Binding energy of electronic clusters | p. 210 |
| Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively | p. 213 |
| Fractors: Fractional Tensor Calculus | p. 219 |
| Covariance for fractional tensors | p. 219 |
| Singular fractional tensors | p. 220 |
| Fractional Fields | p. 223 |
| Fractional Euler-Lagrange equations | p. 224 |
| The fractional Maxwell equations | p. 227 |
| Gauge Invariance in Fractional Field Theories | p. 231 |
| Gauge invariance in first order of the coupling constant g | p. 232 |
| The fractional Riemann-Liouville-Zeeman effect | p. 236 |
| Outlook | p. 241 |
| Bibliography | p. 243 |
| Index | p. 257 |
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