The fractional Laplacian in one dimension Random walkers with constant steps Ordinary diffusion Random jumpers Central limit theorem and stable distributions Power-law probability jump lengths A principal-value integral Wires and springs The fractional Laplacian Fourier transform Effect of fractional order Numerical computation of the fractional Laplacian Green's function of the fractional Laplace equation Fractional Poisson equation in a restricted domain Green's function of unsteady fractional diffusion Numerical discretization in one dimension Computation of a principal-value integral Fractional Laplacian differentiation matrix Fractional Poisson equation Evolution under fractional diffusion Differentiation by spectral expansion Further concepts in one dimension Fractional first derivative Properties of the fractional first derivative Laplacian potential Fractional derivatives from finite-difference stencils Fractional third derivative Fractional fourth derivative Periodic functions Sine, cosines, and the complete Fourier series Cosine Fourier series Sine Fourier series Green's functions Integral representation of the periodic Laplacian Numerical discretization Periodic differentiation matrix Differentiation by spectral expansion Embedding of the fractional Poisson equation The fractional Laplacian in three dimensions Stipulation on the Fourier transform Integral representation Fractional gradient Laplacian potential Green's function of the fractional Laplace equation The Riesz potential Triply periodic Green's function Fractional Poisson equation Evolution under fractional diffusion Periodic functions and arbitrary domains Fractional Stokes flow The fractional Laplacian in two dimensions Stipulation on the Fourier transform Integral representation Fractional gradient Laplacian potential Green's function of the fractional Laplace equation The Riesz potential Doubly periodic Green's function Fractional Poisson equation Evolution due to fractional diffusion Periodic functions and arbitrary domains Appendix A: Selected definite integrals Appendix B: The Gamma function Appendix C: The Gaussian distribution Appendix D: The fractional Laplacian in arbitrary dimensions Appendix E: Fractional derivatives Appendix F: Aitken extrapolation of an infinite sum References Index