  The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local
elliptic operator. While analytical theory has been advanced and understood for
some time, there remain many open problems in the numerical analysis of the
operator. In this article, we study several different finite difference
discretisations of the fractional Laplacian on uniform grids in one dimension
that takes the same form. Many properties can be compared and summarised in
this relatively simple setting, to tackle more important questions like the
nonlocality, singularity and flat tails common in practical implementations.
The accuracy and the asymptotic behaviours of the methods are also studied,
together with treatment of the far field boundary conditions, providing a
unified perspective on the further development of the scheme in higher
dimensions.
