Fractional calculus: an introduction for physicists

Fractional calculus is undergoing rapid and ongoing development. We can already recognize that within its framework, new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics. This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory.

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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 -- 4. Friction forces. 4.1. Classical description. 4.2. Fractional friction -- 5. Fractional calculus. 5.1. The Fourier transform. 5.2. The fractional integral. 5.3. Correlation of fractional integration and differentiation. 5.4. Fractional derivative of second order. 5.5. Fractional derivatives of higher orders. 5.6. Geometric interpretation of the fractional integral. 5.7. Low level fractionality. 5.8. Discussion -- 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 effec…