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We consider a multilayer stripline structure excited by a delta-gap source and we study the continuous-spectrum current, which may be responsible of several spurious transmission effects such as interference with the quasi-TEM mode, undesired radiative effects, and crosstalk. In particular, a uniform asymptotic analysis of such continuous spectrum is presented, based on two alternative closed-form representations. Both representations are accurate also very close to the source and are valid inside a wide frequency range starting from DC. One formulation has a direct implementation, since it needs only knowledge of the spectral Green's function of the background structure. The other formulation requires knowledge of the exact complex (leaky) poles of the stripline Green's function and of their residues, but provides more physical insight since it gives quantitative information on the role of leaky waves in the continuous-spectrum representation in the entire considered frequency range. Numerical results which validate the accuracy and efficiency of the proposed approaches are provided at different frequencies for covered-microstrip structures with different strip widths.