Editing WKB approximation (section)

=== 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.