resummable

Resummable refers to the property of a formal series or asymptotic expansion that can be assigned a finite, well-defined value through a resummation procedure. In analysis and mathematical physics, many series that arise from perturbation theory are divergent, yet can be given meaning by resummation techniques. A resummable object is one for which a consistent procedure produces a function whose behavior matches the original series in the intended limit.

One prominent framework is Borel resummation. For a formal power series sum a_n z^n, the Borel transform B(t) = sum a_n t^n / n! is constructed. If B(t) extends analytically to a suitable domain and grows at most moderately, then a Laplace-type integral can be used to recover a function S(z) that agrees with the series to all orders as z approaches zero. When such a procedure exists, the original series is said to be Borel summable, and S(z) is the resummed value. Resummability may depend on the direction in the complex plane, leading to phenomena like Stokes lines and sectorial behavior.

Beyond Borel resummation there are generalized methods, including lateral and median resummations, and the broader theory of resurgence and transseries. These frameworks study how different regions (sectors) in the complex plane relate, how singularities influence continuation, and how the resummed values capture nonperturbative information.

Applications of resummable expansions appear in perturbative quantum field theory, statistical mechanics, and dynamical systems, where divergent series frequently encode meaningful asymptotics.