  In this article we derive longitudinal impedance and wake function for an
undulator setup with arbitrary undulator parameter, taking into account a
finite transverse size of the electron bunch. Earlier studies considered a line
density-distribution of electrons instead. We focus our attention on the
long-wavelength asymptote (compared with resonance wavelength), at large
distance of the electron bunch from the undulator entrance compared to the
overtaking length, and for large vacuum-chamber size compared to the typical
transverse size of the field. These restrictions define a parameter region of
interest for practical applications. We calculate a closed expression for
impedance and wake function that may be evaluated numerically in the most
general case. Such expression allows us to derive an analytical solution for a
Gaussian transverse and longitudinal bunch shape. Finally, we study the
feasibility of current-enhanced SASE schemes (ESASE) recently proposed for
LCLS, that fall well-within our approximations. Numerical estimations presented
in this paper indicate that impedance-induced energy spread is sufficient to
seriously degrade the FEL performance. Our conclusion is in contrast with
results in literature, where wake calculations for the LCLS case are given in
free-space, as if the presence of the undulator were negligible.
