  A method of calculating Feynman diagrams from their small momentum expansion
[1] is extended to diagrams with zero mass thresholds. We start from the
asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$
are large compared to all momenta squared). Using dimensional regularization, a
finite result is obtained in terms of powers of logarithms (describing the
zero-threshold singularity) times power series in the momentum squared.
Surprisingly, these latter ones represent functions, which not only have the
expected physical `second threshold' but have a branchcut singularity as well
below threshold at a mirror position. These can be understood as
pseudothresholds corresponding to solutions of the Landau equations. In the
spacelike region the imaginary parts from the various contributions cancel. For
the two-loop examples with one mass $M$, in the timelike region for $q^2
\approx M^2$ we obtain approximations of high precision. This will be of
relevance in particular for the calculation of the decay $Z \to b\bar{b}$ in
the $m_b=0$ approximation.
