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In [[mathematics]] and [[physics]], a '''non-perturbative''' [[function (mathematics)|function]] or process is one that cannot be accurately described by [[perturbation theory]]. An example is the function
 
<math>f(x)=e^{-1/x^2}</math>.
 
[[Image:expinvsq.svg|frame|The function ''e''<sup>-1/''x''²</sup>. The Taylor series is identically zero, but the function is not.]]
 
The [[Taylor series]] at x = 0 for this function is exactly zero to all orders in perturbation theory, but the function is non-zero if ''x'' ≠ 0.
 
The implication of this for physics is that there are some phenomena which are impossible to understand by perturbation theory, regardless of how many orders of perturbation theory we use. [[Instanton]]s are an example.
 
Therefore, in [[theoretical physics]], a '''non-perturbative''' solution or theory is one that does not require perturbation theory to explicate, or does not simply describe the dynamics of perturbations around some fixed background.  For this reason, non-perturbative solutions and theories yield insights into areas and subjects perturbative methods cannot reveal.
==See also==
*[[Taylor series]]
*[[Soliton]]
*[[Sphaleron]]
*[[Instanton]]
*[[Loop quantum gravity]]
*[[Causal dynamical triangulation]]
 
[[Category:Perturbation theory]]
 
 
{{maths-stub}}
{{physics-stub}}

Revision as of 02:01, 4 March 2014



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