  In this paper, we extend G{\'e}rard's formula for the solution of the
Benjamin--Ono equation on the line to square integrable and real valued initial
data. Combined with this formula, we also extend the G{\'e}rard's formula for
the zero dispersion limit of the Benjamin--Ono equation on the line to more
singular initial data. In the derivation of the extension of the formula for
the zero dispersion limit, we also find an interesting integral equality, which
might be useful in other contexts.
