QuasiPeriodicHelmholtz

G = NaiveGreensFunction2D(wavenumber,period,theta)

G(x,ξ;M)

Computes 2D quasi-periodic Greens function for Helmholtz equation with

k = wavenumber(G)

that is quasi-periodic with f([x,y],[ξ,η]) = exp(imkdcos(θ))f([ξ+d,η]) where

d = period(G)
θ = theta(G)

by summation of the free space Greens Function for 2M+1 closest periods

G = TailGreensFunction2D(wavenumber,period,theta)

G(x,ξ;M)

Computes 2D quasi-periodic Greens function for Helmholtz equation with

k = wavenumber(G)

that is quasi-periodic with f(x,y) = exp(imxcos(θ))*f(x+d,y) where

d = period(G)
θ = angle(G)

by summation of the free space Greens Function for 2M-1 closest periods, and approximating the tail asymptotically to order M^{-5/2}