Dirichlet kernel

The second result makes use of the Dirichlet kernel, D x) and is given here without proof. 

Dirichlet function, which is an approximation of Delta function, S x). Various approximate representations of Dirac delta function are provided in Van der Pol Bremmer (1959) on pp 61-62. This clearly shows that we recover the applied boundary condition at y = 0. Therefore, the delta function is totally supported by the point at infinity in the wave number space (which is nothing but the circular arc of Fig. 2.20 i.e. the essential singularity of the kernel of the contour integral). 

Ho GS, Ligneres VL, Carter EA Analytic form for a nonlocal kinetic energy functional with a density-dependent kernel for orbital-free density functional theory under periodic and Dirichlet boundary conditions, Phys Rev B 78(4) 045105, 2008. 

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