  We summarize recent progress in applying the worldline formalism to the
analytic calculation of one-loop N-point amplitudes. This string-inspired
approach is well-adapted to avoiding some of the calculational inefficiencies
of the standard Feynman diagram approach, most notably by providing master
formulas that sum over diagrams differing only by the position of external legs
and/or internal propagators. We illustrate the mathematical challenge involved
with the low-energy limit of the N-photon amplitudes in scalar and spinor QED,
and then present an algorithm that, in principle, solves this problem for the
much more difficult case of the N-point amplitudes at full momentum in phi^3
theory. The method is based on the algebra of inverse derivatives in the
Hilbert space of periodic functions orthogonal to the constant ones, in which
the Bernoulli numbers and polynomials play a central role.
