Infinite-order accuracy limit of finite difference formulas in the complex plane

Abstract

Abstract It was recently found that finite difference (FD) formulas become remarkably accurate when approximating derivatives of analytic functions $f(z)$ in the complex $z=x+\text{i}y$ plane. On unit-spaced grids in the $x,y$-plane, the FD weights decrease to zero with the distance to the stencil center at a rate similar to that of a Gaussian, typically falling below the level of double precision accuracy $\mathcal{O}(10^{-16})$ already about four node spacings away from the center point. We follow up on these observations here by analyzing and illustrating the features of such FD stencils in their infinite-order accurate limit (for traditional FD approximations known as their pseudospectral limit).