Logarithmic singularity subtraction for matrix-friendly layered medium Green's functions

A subtraction procedure is proposed to evaluate the logarithmic singularity for matrix-friendly layered medium Green's functions. In this subtraction procedure, an integral of the Bessel function combined with the exponential function and power function, where the sum of the power of k(rho) and the order of the Bessel function is -1, is designed to fit the oscillation of the tail integral. After subtracting the designed integral, the integrand of the remainder decreases rapidly. Accordingly, the remainder can be calculated without finding any surface poles or steepest descent path, while the subtracted part can be calculated analytically. Several numerical examples are provided to reveal the feasibility and the accuracy of the proposed method.