The coherently emitted electric field pulse of a short electron bunch is obtained by summing the fields of the individual electrons, taking phase differences due to different longitudinal positions into account. For an electron density, this sum becomes an integral over the charge density and frequency spectrum of the emitted radiation, which, however, is difficult to evaluate numerically. In this paper, we present a fast analytic method valid for arbitrary bunch shapes. We also include

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... g effects of the beam pipe and consider ultra-short bunches, where the high frequency part of the coherent synchrotron spectrum is cut-off not by the inverse bunch length but by the critical frequency of synchrotron radiation. Our technique is applied to bunches, simulated simulated for the linac-based FLUTE accelerator test facility at KIT.