Contents -- Foreword -- Acknowledgments -- 1. Introduction -- 2. Functions -- 2.1 Gamma function -- 2.2 Mittag-Leffler functions -- 2.3 Hypergeometric functions -- 2.4 Miscellaneous functions -- 3. The Fractional Derivative -- 3.1 Basics -- 3.2 The fractional Leibniz product rule -- 3.3 Discussion -- 3.3.1 Orthogonal polynomials -- 3.3.2 Differential representation of the Riemann fractional derivative -- 4. Friction Forces -- 4.1 Classical description -- 4.2 Fractional friction -- 5. Fractional Calculus -- 5.1 The Fourier transform -- 5.2 The fractional integral -- 5.2.1 The Liouville fractional integral -- 5.2.2 The Riemann fractional integral -- 5.3 Correlation of fractional integration and differentiation -- 5.3.1 The Liouville fractional derivative -- 5.3.2 The Riemann fractional derivative -- 5.3.3 The Liouville fractional derivative with inverted operator sequence: the Liouville-Caputo fractional derivative -- 5.3.4 The Riemann fractional derivative with inverted operator sequence: the Caputo fractional derivative -- 5.4 Fractional derivative of second order -- 5.4.1 The Riesz fractional derivative -- 5.4.2 The Feller fractional derivative -- 5.5 Fractional derivatives of higher orders -- 5.6 Geometric interpretation of the fractional integral -- 5.7 Low level fractionality -- 5.8 Discussion -- 5.8.1 Semigroup property of the fractional integral -- 6. The Fractional Harmonic Oscillator -- 6.1 The fractional harmonic oscillator -- 6.2 The harmonic oscillator according to Fourier -- 6.3 The harmonic oscillator according to Riemann -- 6.4 The harmonic oscillator according to Caputo -- 7. Wave Equations and Parity -- 7.1 Fractional wave equations -- 7.2 Parity and time-reversal -- 7.3 Solutions of the free regularized fractional wave equation -- 8. Nonlocality and Memory Effects -- 8.1 A short history of nonlocal concepts.

8.2 From local to nonlocal operators -- 8.3 Memory effects -- 9. Quantum Mechanics -- 9.1 Canonical quantization -- 9.2 Quantization of the classical Hamilton function and free solutions -- 9.3 Temperature dependence of a fission yield and determination of the corresponding fission potential -- 9.4 The fractional Schrödinger equation with an infinite well potential -- 9.5 Radial solutions of the fractional Schrödinger equation -- 10. Fractional Spin: a Property of Particles Described with the Fractional Schrödinger Equation -- 10.1 Spin -- 10.2 Fractional spin -- 11. Factorization -- 11.1 The Dirac equation -- 11.2 The fractional Dirac equation -- 11.3 The fractional Pauli equation -- 12. Symmetries -- 12.1 Characteristics of fractional group theory -- 12.2 The fractional rotation group SO N -- 13. The Fractional Symmetric Rigid Rotor -- 13.1 The spectrum of the fractional symmetric rigid rotor -- 13.2 Rotational limit -- 13.3 Vibrational limit -- 13.4 Davidson potential: the so called -unstable limit -- 13.5 Linear potential limit -- 13.6 The magic limit -- 13.7 Comparison with experimental data -- 14. q-deformed Lie Algebras and Fractional Calculus -- 14.1 q-deformed Lie algebras -- 14.2 The fractional q-deformed harmonic oscillator -- 14.3 The fractional q-deformed symmetric rotor -- 14.4 Half-integer representations of the fractional rotation group SO (3) -- 15. Fractional Spectroscopy of Hadrons -- 15.1 Phenomenology of the baryon spectrum -- 15.2 Charmonium -- 15.3 Phenomenology of meson spectra -- 15.4 Metaphysics: About the internal structure of quarks -- 16. Higher Dimensional Fractional Rotation Groups -- 16.1 The four decompositions of the mixed fractional SO (9) -- 16.2 Notation -- 16.3 The nine dimensional fractional Caputo-Riemann-Riemann symmetric rotor -- 16.4 Magic numbers of nuclei -- 16.5 Ground state properties of nuclei.

16.6 Fine structure of the single particle spectrum: the extended Caputo-Riemann-Riemann symmetric rotor -- 16.7 Magic numbers of electronic clusters: the nine dimensional fractional Caputo-Caputo-Riemann symmetric rotor -- 16.8 Binding energy of electronic clusters -- 16.9 Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively -- 17. Fractors: Fractional Tensor Calculus -- 17.1 Covariance for fractional tensors -- 17.2 Singular fractional tensors -- 18. Fractional Fields -- 18.1 Fractional Euler-Lagrange equations -- 18.2 The fractional Maxwell equations -- 19. Gauge Invariance in Fractional Field Theories -- 19.1 Gauge invariance in first order of the coupling constant ḡ -- 19.2 The fractional Riemann-Liouville-Zeeman effect -- 20. Outlook -- Bibliography -- Index.