  Electromagnetic fields induced by the space charge in relativistic beams play
an important role in Accelerator Physics. They lead to emittance growth, slice
energy change, and the microbunching instability. Typically, these effects are
modeled numerically since simple description exists only in the limits of
large- or small-scale current variations. In this paper we consider an axially
symmetric charged beam inside a round pipe and find the solution of the space
charge problem that is valid in the full range of current variations. We
express the solution for the field components in terms of Green's functions,
which are fully determined by just a single function. We then find that this
function is an on-axis potential from a charged disk in a round pipe, with
transverse charge density $\rho_\perp(r)$, and it has a compact analytical
expression. We finally provide an integrated Green's function based approach
for efficient numerical evaluation in the case when the transverse charge
density stays the same along the beam.
