IMPROVEMENT OF WATSON THEOREM ON BOREL SUMMABILITY

Watson's theorem, which gives sufficient conditions for Borel summability, is not optimal. Watson assumes analyticity and uniform asymptotic expansion in a sector |argz|<π/2 + ε, |z| < R, with ε > 0; in fact, only the circular region Re(1/z) > 1/R is required. In particular, one can take ε = 0. This improved theorem gives a necessary and sufficient characterization of a large class of Borel-summable functions. I apply it to the perturbation expansion in the φ24 quantum field theory. © 1980 American Institute of Physics.