It is known until now that when most printed-circuit transmission lines are excited by a practical source of a monochromatic high frequency, the near-field amplitude decays exponentially along the guide axis due to the excitation of a leaky mode. This is true if the excitation frequency is in the leaky portion of the guide and the leaky mode is the only physical mode at this frequency. Then, the decay rate of the field amplitude is given by the leakage constant of the physical leaky mode. However, we have recently found that although the excitation is set at the frequency at which the leaky mode is the only physical solution, there is a situation in which the amplitude decay of the near field becomes much smaller than that expected from the leaky-mode eigenvalue. Then, contrary to our earlier belief that nonphysical solutions exert little influence on the physical field, we have recently discovered that such a slow decay in the physical near-field amplitude is strongly influenced by an improper-real solution, which is in the leakage portion as a nonphysical solution. We report here for the first time that the slow decay in the near field depends on how close the nonphysical improper-real solution is to the surface-wave curve relevant to power leakage. In this paper we present the evidence for this discovery by a discussion based on the generalized pencil of function analysis applied to the finite difference time domain data. To reinforce these discussions, we also present discussions viewed from the mutual relation between the excited-field distributions on the guide and the evolution of the improper-real solution on the steepest-descent plane.