Editing Perturbation theory (section)

{{Short description|Methods of mathematical approximation}}
{{About|perturbation theory as a general mathematical method|perturbation theory applied specifically to quantum mechanics|Perturbation theory (quantum mechanics)}}
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{{Differential equations}}

In [[mathematics]] and [[applied mathematics]], '''perturbation theory''' comprises methods for finding an [[approximation theory|approximate solution]] to a problem, by starting from the exact [[solution (equation)|solution]] of a related, simpler problem.<ref name=":0">{{Cite book|last=Bender|first=Carl M.| url=https://www.worldcat.org/oclc/851704808|title=Advanced mathematical methods for scientists and engineers I : asymptotic methods and perturbation theory|date=1999| others=Steven A. Orszag| isbn=978-1-4757-3069-2|location=New York, NY|oclc=851704808 |publisher=Springer}}</ref><ref name=":1">{{Cite book| last=Holmes|first=Mark H.| url=https://www.worldcat.org/oclc/821883201|title=Introduction to perturbation methods|date=2013| publisher=Springer| isbn=978-1-4614-5477-9|edition=2nd|location=New York| oclc=821883201}}</ref> A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts.<ref name="General Perturbations">{{cite book |isbn= 978-145378-1470|title=Modern Astrodynamics |author=William E. Wiesel |location=Ohio |publisher=Aphelion Press| year=2010 |page=107}}</ref>  In '''regular perturbation theory''', the solution is expressed as a [[power series]] in a small parameter {{nowrap|<math>\varepsilon</math>.}}<ref name=":0" /><ref name=":1" /> The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of <math>\varepsilon</math> usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.

Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in [[quantum field theory]]. [[Perturbation theory (quantum mechanics)]] describes the use of this method in [[quantum mechanics]]. The field in general remains actively and heavily researched across multiple disciplines.