theory of divergent series

E451513

branch of mathematical analysis mathematical theory

The theory of divergent series is a branch of mathematical analysis that studies how to assign meaningful values to infinite series that do not converge in the usual sense, using specialized summation methods and analytic continuation.

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Statements (51)

Predicate	Object
instanceOf	branch of mathematical analysis ⓘ mathematical theory ⓘ
aimsTo	extend the notion of sum of a series ⓘ provide consistent rules for assigning values to divergent series ⓘ relate divergent expansions to analytic functions ⓘ
appliesTo	asymptotic power series in physics ⓘ formal series in differential equations ⓘ generating functions outside their domain of convergence ⓘ perturbation series in quantum field theory ⓘ power series with zero radius of convergence ⓘ
fieldOfStudy	analytic continuation of series ⓘ asymptotic expansions ⓘ divergent series ⓘ generalized summation methods ⓘ resummation techniques ⓘ summation of divergent series ⓘ
hasApplication	definition of path integrals via regularization ⓘ evaluation of asymptotic expansions in applied mathematics ⓘ regularization of divergent integrals ⓘ
hasHistoricalFigure	Ernst Cesàro NERFINISHED ⓘ G. H. Hardy NERFINISHED ⓘ Leonhard Euler NERFINISHED ⓘ Niels Henrik Abel NERFINISHED ⓘ Srinivasa Ramanujan NERFINISHED ⓘ Émile Borel NERFINISHED ⓘ
hasKeyResult	Hardy’s classification of summability methods ⓘ Tauberian theorems relating summability to convergence ⓘ conditions for equivalence of summation methods ⓘ
relatedTo	asymptotic analysis ⓘ complex analysis ⓘ functional analysis ⓘ perturbation theory ⓘ quantum field theory ⓘ special functions ⓘ summability theory ⓘ
studies	behavior of series outside their radius of convergence ⓘ infinite series that do not converge in the usual sense ⓘ methods to assign finite values to divergent series ⓘ relations between different summation methods ⓘ
usesConcept	Abel summation ⓘ Borel summation NERFINISHED ⓘ Borel transform NERFINISHED ⓘ Cesàro summation NERFINISHED ⓘ Euler summation NERFINISHED ⓘ Ramanujan summation NERFINISHED ⓘ Stokes phenomenon NERFINISHED ⓘ analytic continuation ⓘ asymptotic series ⓘ resurgence theory ⓘ summability ⓘ zeta function regularization ⓘ

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Orders of Infinity → hasPart → theory of divergent series ⓘ