  A new derivation is given for the representation, under certain conditions,
of the integral dispersion relations of scattering theory through local forms.
The resulting expressions have been obtained through an independent procedure
to construct the real part, and consist of new mathematical structures of
double infinite summations of derivatives. In this new form the derivatives are
calculated at the generic value of the energy $E$ and separately at the
reference point $E=m$ that is the lower limit of the integration. This new form
may be more interesting in certain circumstances and directly shows the origin
of the difficulties in convergence that were present in the old truncated forms
called standard-DDR. For all cases in which the reductions of the double to
single sums were obtained in our previous work, leading to explicit
demonstration of convergence, these new expressions are seen to be identical to
the previous ones. We present, as a glossary, the most simplified explicit
results for the DDR's in the cases of imaginary amplitudes of forms
$(E/m)^\lambda[\ln (E/m)]^n$, that cover the cases of practical interest in
particle physics phenomenology at high energies. We explicitly study the
expressions for the cases with $\lambda$ negative odd integers, that require
identification of cancelation of singularities, and provide the corresponding
final results.
