The stratified medium fast multipole algorithm (SMFMA) is a fast integral equation method designed for static electromagnetic coupling analysis, such as capacitance extractions. In it the multipole translators are intensively employed to account for the interactions between source and field. However, the multipole translators do not have closed forms in the presence of the multilayer medium. Numerical integrations were used to compute them, which were time consuming. In this paper, the closed forms of the multipole translators are derived by extending the spherical harmonic functions into the complex plane. Using DCIM and combining the complex images of each real image from the multilayer medium, the outgoing to local multipole translator computation can be significantly shortened. One to two orders speedup can be easily achieved with very good accuracy.
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