  The representation of the usual integral dispersion relations (IDR) of
scattering theory through series of derivatives of the amplitudes is discussed,
extended, simplified, and confirmed as mathematical identities. Forms of
derivative dispersion relations (DDR) valid for the whole energy interval,
recently obtained and presented as double infinite series, are simplified
through the use of new sum rules of the incomplete $\Gamma$ functions, being
reduced to single summations, where the usual convergence criteria are easily
applied. For the forms of the imaginary amplitude used in phenomenology of
hadronic scattering, we show that expressions for the DDR can represent, with
absolute accuracy, the IDR of scattering theory, as true mathematical
identities. Besides the fact that the algebraic manipulation can be easily
understood, numerical examples show the accuracy of these representations up to
the maximum available machine precision. As consequence of our work, it is
concluded that the standard simplified forms sDDR, originally intended for the
high energy limits, are an inconvenient and incomplete separation of terms of
the full expression, leading to wrong evaluations. Since the correspondence
between IDR and the DDR expansions is linear, our results have wide
applicability, covering more general functions, built as combinations of well
studied basic forms.
