92 research outputs found
Resurgence of the Euler-MacLaurin summation formula
The Euler-MacLaurin summation formula relates a sum of a function to a
corresponding integral, with a remainder term. The remainder term has an
asymptotic expansion, and for a typical analytic function, it is a divergent
(Gevrey-1) series. Under some decay assumptions of the function in a half-plane
(resp. in the vertical strip containing the summation interval), Hardy (resp.
Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and
give an exact Euler-MacLaurin summation formula.
Using a mild resurgence hypothesis for the function to be summed, we give a
Borel summable transseries expression for the remainder term, as well as a
Laplace integral formula, with an explicit integrand which is a resurgent
function itself. In particular, our summation formula allows for resurgent
functions with singularities in the vertical strip containing the summation
interval.
Finally, we give two applications of our results. One concerns the
construction of solutions of linear difference equations with a small
parameter. And another concerns the problem of proving resurgence of formal
power series associated to knotted objects.Comment: AMS-LaTeX, 15 pages with 2 figure
Boundary blow-up solutions in the unit ball : asymptotics, uniqueness and symmetry
We calculate the full asymptotic expansion of boundary blow-up so- lutions,
for any nonlinearity f . Our approach enables us to state sharp qualitative
results regarding uniqueness and ra- dial symmetry of solutions, as well as a
characterization of nonlinearities for which the blow-up rate is universal. At
last, we study in more detail the standard nonlinearities f (u) = u^p, p >
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