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Statements

Subject Item
dbr:Equichordal_point_problem
dbo:wikiPageWikiLink
dbr:Perturbation_problem_beyond_all_orders
Subject Item
dbr:Perturbation_problem_beyond_all_orders
rdfs:label
Perturbation problem beyond all orders
rdfs:comment
In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by a power series.
dcterms:subject
dbc:Asymptotic_analysis dbc:Perturbation_theory
dbo:wikiPageID
29746108
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1114297657
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dbc:Asymptotic_analysis dbr:Essential_singularity dbr:Asymptotic_expansion dbr:Flat_function dbr:Power_series dbc:Perturbation_theory dbr:Perturbation_theory dbr:Radius_of_convergence dbr:Laurent_series dbr:Taylor_series
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dbp:date
2017-01-09
dbp:url
n10:12-psi-mathematical-physics
dbo:abstract
In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by a power series. A simple example is understood by an attempt at trying to expand in a Taylor series in about 0. All terms in a naïve Taylor expansion are identically zero. This is because the function possesses an essential singularity at in the complex -plane, and therefore the function is most appropriately modeled by a Laurent series -- a Taylor series has a zero radius of convergence. Thus, if a physical problem possesses a solution of this nature, possibly in addition to an analytic part that may be modeled by a power series, the perturbative analysis fails to recover the singular part. Terms of nature similar to are considered to be "beyond all orders" of the standard perturbative power series.
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