  We write the Green function of the $d$-dimensional hypercubic lattice in a
piecewise form covering the entire real frequency axis. Each piece is a single
integral involving modified Bessel functions of the first and second kinds. The
smoothness of the integrand allows both real and imaginary parts of the Green
function to be computed quickly and accurately for any dimension $d$ and any
real frequency, and the computational time scales only linearly with $d$.
