  Based on a study of the properties of the Lerch's transcendent, exact closed
forms of dispersion relations for amplitudes and for derivatives of amplitudes
in pp and p\=p scattering are introduced. Exact and complete expressions are
written for the real parts and for their derivatives at $t=0$ based on given
inputs for the energy dependence of the total cross sections and of the slopes
of the imaginary parts. The results are prepared for application in the
analysis of forward scattering data of the pp and p\=p systems at all energies,
where exact and precise representations can be written.
