Mathematics – Classical Analysis and ODEs
Scientific paper
2006-08-16
Proc. R. Soc. Lond. A 455, 1931-1956 (1999)
Mathematics
Classical Analysis and ODEs
Scientific paper
We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions {\em not} excepted) there exists a nonempty set of solutions whose difference from the ``optimally'' (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas $\mathcal{B}$ which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, $\mathcal{B}_0$, such that for any formal series solution $\tilde{f}$, $\mathcal{B}_0(\tilde{f})$ differs from the optimal truncation of $\tilde{f}$ by at most the order of the least term of $\tilde{f}$. We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms ``beyond all orders'' {\em change} in crossing a Stokes line, these terms vary smoothly on the Berry scale $\arg(x)\sim |x|^{-1/2}$ and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds.
Costin Ovidiu
Kruskal Martin D.
No associations
LandOfFree
On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-220484