  We discuss some formal and fundamental aspects related with the replacement
of integral dispersion relations by derivative forms, and their practical uses
in high energy elastic hadron scattering, in particular $pp$ and $\bar{p}p$
scattering. Starting with integral relations with one subtraction and
considering parametrizations for the total cross sections belonging to the
class of entire functions in the logarithm of the energy, a series of results
is deduced and our main conclusions are the following: (1) except for the
subtraction constant, the derivative forms do not depend on any additional free
parameter; (2) the only approximation in going from integral to derivative
relations (at high energies) concerns to assume as zero the lower limit in the
integral form; (3) the previous approximation and the subtraction constant
affect the fit results at both low and high energies and therefore, the
subtraction constant can not be disregarded; (4) from a practical point of
view, for single-pole Pomeron and secondary reggeons parametrizations and
center-of-mass energies above 5 GeV, the derivative relations with the
subtraction constant as a free fit parameter are completely equivalent to the
integral forms with finite (non-zero) lower limit. A detailed review on the
conditions of validity and assumptions related with the replacement of integral
by derivative relations is also presented and discussed.

High Energy Physics - Phenomenology;High Energy Physics - Experiment;High Energy Physics - Theory