This work proposes a semi-analytical method for simulation of the acoustic response of multipole eccentered sources in a fluid-filled borehole. Assuming a geometry that is invariant with respect to the azimuthal and vertical directions, the solution in frequency domain is expressed in terms of a Fourier series and a Fourier integral. The proposed semi-analytical method builds upon the idea of separating singularities from the smooth part of the integrand when performing the inverse Fourier transform. The singular part is treated analytically using existing inversion formulae, while the regular part is treated with a FFT technique. As a result, a simple and effective method that can be used for simulating and understanding the main physical principles occurring in borehole-eccentered sonic measurements is obtained. Numerical results verify the proposed method and illustrate its advantages.