Perturbation problem beyond all orders

In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by a power series. A simple example is understood by an attempt at trying to expand e − 1 / ϵ {\displaystyle e^{-1/\epsilon }} in a Taylor series in ϵ > 0 {\displaystyle \epsilon >0} about 0. All terms in a naïve Taylor expansion are identically zero. This is because the function e − 1 / z {\displaystyle e^{-1/z}} possesses an essential singularity at z = 0 {\displaystyle z=0} in the complex z {\displaystyle z} -plane, and therefore the function is most appropriately modeled by a Laurent series -- a Taylor series has a zero radius of convergence. Thus, if a physical problem possesses a solution of this nature, possibly in addition to an analytic part that may be modeled by a power series, the perturbative analysis fails to recover the singular part. Terms of nature similar to e − 1 / ϵ {\displaystyle e^{-1/\epsilon }} are considered to be "beyond all orders" of the standard perturbative power series.

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