  A new powerful method to calculate Feynman diagrams is proposed. It consists
in setting up a Taylor series expansion in the external momenta squared (in
general multivariable). The Taylor coefficients are obtained from the original
diagram by differentiation and putting the external momenta equal to zero,
which means a great simplification. It is demonstrated that it is possible to
obtain by analytic continuation of the original series high precision numerical
values of the Feynman integrals in the whole cut plane. For this purpose
conformal mapping and subsequent resummation by means of Pad\'{e} approximants
or Levin transformation are applied.
