Exact treatment of dispersion relations in pp and p\=p elastic scattering

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.