Editing WKB approximation (section)

==Formulation==

Generally, WKB theory is a method for approximating the solution of a differential equation whose ''highest derivative is multiplied by a small parameter'' {{mvar|ε}}. The method of approximation is as follows.

For a differential equation
<math display="block"> \varepsilon \frac{d^ny}{dx^n} + a(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + k(x)\frac{dy}{dx} + m(x)y= 0,</math>
assume a solution of the form of an [[asymptotic series]] expansion
<math display="block"> y(x) \sim \exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty} \delta^n S_n(x)\right]</math>
in the limit {{math|''δ'' → 0}}. The asymptotic scaling of {{mvar|δ}} in terms of {{mvar|ε}} will be determined by the equation – see the example below.

Substituting the above [[ansatz]] into the differential equation and cancelling out the exponential terms allows one to solve for an arbitrary number of terms {{math|''S''<sub>''n''</sub>(''x'')}} in the expansion.

WKB theory is a special case of [[multiple scale analysis]].<ref>{{cite book
 | title = Acoustics: basic physics, theory and methods
 | first = Paul 
 | last = Filippi
 | publisher = Academic Press
 | year = 1999
 | isbn = 978-0-12-256190-0
 | page = 171
 | url = https://books.google.com/books?id=xHWiOMp63WsC&q=wkb%20multi-scale&pg=PA171
 }}</ref><ref>
{{Cite book 
 | author1=Holmes, M. 
 | title=Introduction to Perturbation Methods, 2nd Ed 
 | year=2013 
 | publisher=Springer 
 | isbn=978-1-4614-5476-2 
 }}</ref><ref name=":0">{{cite book
 | first1=Carl M.
 | last1=Bender
 | author-link1=Carl M. Bender
 | first2=Steven A.
 | last2=Orszag
 | author-link2=Steven Orszag
 | title=Advanced mathematical methods for scientists and engineers
 | publisher=Springer
 | year=1999
 | isbn=0-387-98931-5
 | pages=549–568
 }}</ref>