  Growing evidence indicates that the synchrotron radiation mechanism may be
responsible for the prompt emission of gamma-ray bursts (GRBs). In the
synchrotron radiation scenario, the electron energy spectrum of the prompt
emission is diverse in theoretical works and has not been estimated from
observations in a general way (i.e., without specifying a certain physical
model for the electron spectrum). In this paper, we creatively propose a method
to directly estimate the electron spectrum for the prompt emission, without
specifying a certain physical model for the electron spectrum in the
synchrotron radiation scenario. In this method, an empirical function (i.e., a
four-order Bezier curve jointed with a linear function at high-energy) is
applied to describe the electron spectrum in log-log coordinate. It is found
that our empirical function can well mimic the electron spectra obtained in
many numerical calculations or simulations. Then, our method can figure out the
electron spectrum for the prompt emission without specifying a model. By
employing our method on observations, taking GRB 180720B and GRB 160509A as
examples, it is found that the obtained electron spectra are generally
different from that in the standard fast-cooling scenario and even a broken
power law. Moreover, the morphology of electron spectra in its low-energy
regime varies with time in a burst and even in a pulse. Our proposed method
provides a valuable way to confront the synchrotron radiation mechanism with
observations.
