resummations

Resummation is a collection of mathematical techniques used to assign finite values to series that are divergent or only slowly convergent, and to extract meaningful information from such series. It is widely employed in physics and applied mathematics, where perturbative expansions often produce asymptotic series that do not converge but still encode useful data about the function they approximate.

A central idea is to transform the original series into a form more amenable to summation or analytic continuation. Borel resummation is a foundational method: for a series sum a_n ε^n, one forms the Borel transform B(t) = sum a_n t^n / n!, and then if B(t) can be analytically continued and the integral S(ε) = ∫_0^∞ e^{-t} B(ε t) dt exists, S(ε) is taken as the resummed value. When B(t) has singularities along the integration path, modified procedures such as principal-value prescriptions or combinations with Padé approximants (Borel-Padé) are used to achieve analytic continuation.

Padé approximants provide rational-function approximations to truncated series and can be combined with Borel transforms to improve convergence and continuation. Conformal mapping further enhances summation by transforming the Borel plane to control singularities and expand analyticity domains. Other techniques include Cesàro and Abel summation, Euler transformations, hypergeometric resummation, and optimized or variational perturbation methods.

Applications span quantum field theory, statistical mechanics, critical phenomena, lattice gauge theory, and fluid dynamics, where resummation helps relate perturbative predictions to observable quantities. Limitations include dependence on the chosen method, sensitivity to the available data, and cases where no reliable analytic continuation exists or results are not unique.