Editing WKB approximation (section)

==An example==
This example comes from the text of [[Carl M. Bender]] and [[Steven Orszag]].<ref name=":0" /> Consider the second-order homogeneous linear differential equation
<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>
where <math>Q(x) \neq 0</math>. Substituting
<math display="block">y(x) = \exp \left[\frac{1}{\delta} \sum_{n=0}^\infty \delta^n S_{n}(x)\right]</math>
results in the equation
<math display="block">\epsilon^2\left[\frac{1}{\delta^2} \left(\sum_{n=0}^\infty \delta^nS_{n}^{\prime}\right)^2 + \frac{1}{\delta} \sum_{n=0}^{\infty}\delta^n S_{n}^{\prime\prime}\right] = Q(x).</math>

To [[leading-order|leading order]] in  ''ϵ'' (assuming, for the moment, the series will be asymptotically consistent), the above can be approximated as
<math display="block">\frac{\epsilon^2}{\delta^2} {S_{0}^{\prime}}^2 + \frac{2\epsilon^2}{\delta} S_{0}^{\prime} S_{1}^{\prime} + \frac{\epsilon^2}{\delta} S_{0}^{\prime\prime} = Q(x).</math>

In the limit {{math|''δ'' → 0}}, the [[Method of dominant balance|dominant balance]] is given by
<math display="block">\frac{\epsilon^2}{\delta^2} {S_{0}^{\prime}}^2 \sim Q(x).</math>

So {{mvar|δ}} is proportional to ''ϵ''. Setting them equal and comparing powers yields
<math display="block">\epsilon^0: \quad {S_{0}^{\prime}}^2 = Q(x),</math>
which can be recognized as the [[eikonal equation]], with solution
<math display="block">S_{0}(x) = \pm \int_{x_0}^x \sqrt{Q(x')}\,dx'.</math>

Considering first-order powers of  {{mvar|ϵ}} fixes
<math display="block">\epsilon^1: \quad 2 S_{0}^{\prime} S_{1}^{\prime} + S_{0}^{\prime\prime} = 0.</math>
This has the solution
<math display="block">S_{1}(x) = -\frac{1}{4} \ln Q(x) + k_1,</math>
where {{math|''k''<sub>1</sub>}} is an arbitrary constant.

We now have a pair of approximations to the system (a pair, because   {{math|''S''<sub>0</sub>}} can take two signs); the first-order WKB-approximation will be a linear combination of the two:
<math display="block">y(x) \approx c_1 Q^{-\frac{1}{4}}(x) \exp\left[\frac{1}{\epsilon} \int_{x_0}^x \sqrt{Q(t)} \, dt\right] + c_2 Q^{-\frac{1}{4}}(x) \exp\left[-\frac{1}{\epsilon} \int_{x_0}^x\sqrt{Q(t)} \, dt\right].</math>

Higher-order terms can be obtained by looking at equations for higher powers of {{mvar|δ}}.  Explicitly,
<math display="block"> 2S_{0}^{\prime} S_{n}^{\prime} + S^{\prime\prime}_{n-1} + \sum_{j=1}^{n-1}S^{\prime}_{j} S^{\prime}_{n-j} = 0</math>
for {{math|''n'' ≥ 2}}.

=== Precision of the asymptotic series ===
The asymptotic series for {{math|''y''(''x'')}} is usually a [[divergent series]], whose general term {{math|''δ''<sup>''n''</sup> ''S''<sub>''n''</sub>(''x'')}} starts to increase after a certain value {{math|1=''n'' = ''n''<sub>max</sub>}}. Therefore, the smallest error achieved by the WKB method is at best of the order of the last included term.

For the equation
<math display="block"> \epsilon^2 \frac{d^2 y}{dx^2} = Q(x) y, </math>
with {{math|''Q''(''x'') <0}} an analytic function, the value <math>n_\max</math> and the magnitude of the last term can be estimated as follows:<ref>{{cite journal| last=Winitzki |first=S. |year=2005 |arxiv=gr-qc/0510001 |title=Cosmological particle production and the precision of the WKB approximation |journal=Phys. Rev. D |volume=72 |issue=10 |pages=104011, 14&nbsp;pp |doi=10.1103/PhysRevD.72.104011 |bibcode = 2005PhRvD..72j4011W |s2cid=119152049 }}</ref>
<math display="block">n_\max \approx 2\epsilon^{-1} \left| \int_{x_0}^{x_{\ast}} \sqrt{-Q(z)}\,dz \right| , </math>
<math display="block">\delta^{n_\max}S_{n_\max}(x_0) \approx \sqrt{\frac{2\pi}{n_\max}} \exp[-n_\max], </math>
where <math>x_0</math> is the point at which <math>y(x_0)</math> needs to be evaluated and <math>x_{\ast}</math> is the (complex) turning point where <math>Q(x_{\ast}) = 0</math>, closest to <math>x = x_0</math>.

The number {{math|''n''<sub>max</sub>}} can be interpreted as the number of oscillations between <math>x_0</math> and the closest turning point.

If <math>\epsilon^{-1}Q(x)</math> is a slowly changing function,
<math display="block">\epsilon\left| \frac{dQ}{dx} \right| \ll Q^2 , ^{\text{[might be }Q^{3/2}\text{?]}}</math>
the number {{math|''n''<sub>max</sub>}}  will be large, and the minimum error of the asymptotic series will be exponentially small.