Borel summation
E451516
Borel summation is a mathematical technique that assigns finite values to certain divergent series by transforming and analytically continuing their associated power series.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical technique
ⓘ
method of summability ⓘ summation method ⓘ |
| appliesTo |
asymptotic series
ⓘ
divergent power series ⓘ formal power series ⓘ |
| basedOn |
Borel transform
ⓘ
Laplace transform ⓘ |
| characterizedBy |
analytic continuation of Borel transform
ⓘ
integral representation ⓘ use of exponential damping factor ⓘ |
| compatibleWith | ordinary convergence of series ⓘ |
| field |
asymptotic analysis
ⓘ
complex analysis ⓘ functional analysis ⓘ mathematical analysis ⓘ summability theory ⓘ |
| generalizes | ordinary summation of convergent series ⓘ |
| hasVariant |
Borel–Leroy summation
NERFINISHED
ⓘ
Borel–Écalle summation NERFINISHED ⓘ iterated Borel summation ⓘ |
| historicalDevelopment |
developed by Émile Borel in the context of divergent series
ⓘ
introduced in early 20th century ⓘ |
| namedAfter | Émile Borel NERFINISHED ⓘ |
| property |
can assign finite values to some factorially divergent series
ⓘ
linear summation method ⓘ regular summation method on convergent series ⓘ |
| relatedTo |
Laplace–Borel transform
NERFINISHED
ⓘ
Stokes phenomena NERFINISHED ⓘ Watson’s lemma NERFINISHED ⓘ resummation methods ⓘ Écalle resurgence ⓘ |
| requires |
analytic continuation along integration path
ⓘ
existence of Borel transform ⓘ suitable growth conditions at infinity ⓘ |
| strongerThan |
Abel summation
ⓘ
Cesàro summation ⓘ |
| usedFor |
analytic continuation of power series
ⓘ
assigning values to divergent series ⓘ regularization of divergent series ⓘ resummation of asymptotic expansions ⓘ |
| usedIn |
analytic number theory
ⓘ
differential equations ⓘ perturbation theory ⓘ quantum field theory ⓘ resurgence theory ⓘ theory of divergent integrals ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.