The Stokes phenomenon is a behavior observed in the asymptotic analysis of complex‑valued functions, wherein the dominant terms of an asymptotic expansion change discontinuously as the argument of the function crosses certain lines, known as Stokes lines, in the complex plane. The effect is named after the 19th‑century mathematician Sir George Gabriel Stokes, who first described it in the context of the asymptotic expansion of the Airy function.
Definition
Consider a function $f(z)$ that admits a formal asymptotic expansion
$$ f(z) \sim \sum_{n=0}^{\infty} a_n(z) , e^{\phi_n(z)}, \qquad |z|\to\infty , $$
where each $\phi_n(z)$ is a complex phase function and the $a_n(z)$ are algebraic prefactors. The Stokes phenomenon refers to the situation in which the contribution of a particular exponential term $e^{\phi_k(z)}$ to the asymptotic series is exponentially small on one side of a curve in the complex $z$-plane, but becomes of comparable magnitude (or dominant) on the other side. The curves along which this switching occurs are called Stokes lines (or Stokes rays). The complementary set of curves where the exponentially small terms cancel is termed the anti‑Stokes lines.
Mathematical Characterization
Let $S$ be a Stokes line emanating from a singular point $z_0$. Along $S$ the real part of the exponent $\phi_k(z)$ satisfies
$$ \operatorname{Re},\phi_k(z) = \operatorname{Re},\phi_j(z) $$
for two competing exponential contributions $e^{\phi_k(z)}$ and $e^{\phi_j(z)}$. Crossing $S$ alters the Stokes multiplier—a coefficient that multiplies the exponentially small term in the asymptotic representation. Formally, if $f(z)$ has the representation
$$ f(z) \sim \sum_{n} A_n(z) e^{\phi_n(z)} \quad (\text{in a sector}), $$
then after crossing a Stokes line the coefficients $A_n(z)$ are transformed according to a Stokes matrix, which is typically unipotent (identity plus a rank‑one modification). The change is generally abrupt but analytic continuation of the exact function $f(z)$ remains continuous.
Historical Development
- Sir G. G. Stokes (1857–1903): While studying the asymptotics of the Airy integral, Stokes identified the sudden appearance of an additional exponential term when the argument passed a certain direction in the complex plane.
- J. B. B. W. H. Olver (1974) and C. J. Howls (1992) expanded the theory, formalizing Stokes lines and Stokes matrices within the framework of hyperasymptotics and resurgence.
- The concept has since been incorporated into the theory of Borel summation, exact WKB analysis, and resurgent trans‑series.
Representative Examples
| Function | Asymptotic Form (large $|z|$) | Stokes Lines | |----------|--------------------------------|--------------| | Airy function $\operatorname{Ai}(z)$ | $\displaystyle \operatorname{Ai}(z) \sim \frac{1}{2\sqrt{\pi}} z^{-1/4} e^{-\frac{2}{3}z^{3/2}}$ (for $|\arg z|<\pi$) | $\arg z = \pm \tfrac{2\pi}{3}$ | | Bessel function $K_ u(z)$ | $\displaystyle K_ u(z) \sim \sqrt{\frac{\pi}{2z}} e^{-z}$ | $\arg z = \pm \pi$ | | Exponential integral $\operatorname{Ei}(z)$ | $\displaystyle \operatorname{Ei}(z) \sim \frac{e^{z}}{z}\bigl[1+\frac{1!}{z}+\frac{2!}{z^{2}}+\cdots\bigr]$ | $\arg z = \pi$ (negative real axis) |
In each case, the dominant exponential term switches across the listed Stokes lines, and an exponentially small contribution becomes visible or invisible depending on the side of the line.
Applications
- Quantum Mechanics – In semiclassical (WKB) approximations, the Stokes phenomenon governs tunnelling amplitudes and connection formulas between classically allowed and forbidden regions.
- Fluid Dynamics – Stokes lines appear in the asymptotics of solutions to the Navier–Stokes equations near singularities or high‑Reynolds‑number limits.
- Special Functions – Precise error bounds for asymptotic expansions of Bessel, Airy, parabolic cylinder, and hypergeometric functions rely on understanding Stokes switching.
- Resurgence Theory – The phenomenon is a concrete manifestation of the resurgent structure of divergent series, where Stokes matrices encode the monodromy of the Borel plane.
Formalism in Exact WKB
For a linear differential equation with a large parameter $\hbar$,
$$ \hbar^2 \frac{d^2\psi}{dx^2}=Q(x)\psi, $$
the WKB solutions are of the form
$$ \psi_{\pm}(x) \sim \frac{1}{\sqrt{p(x)}}\exp!\Bigl(\pm\frac{1}{\hbar}\int^x p(t),dt\Bigr), \qquad p(x)=\sqrt{Q(x)}. $$
Stokes lines are defined by $\operatorname{Im}!\bigl(\int^x p(t),dt\bigr)=0$. Across these lines the connection formula mixes $\psi_{+}$ and $\psi_{-}$ via a Stokes matrix
$$ \begin{pmatrix}\psi_{+}\ \psi_{-}\end{pmatrix}_{\text{new}}
\begin{pmatrix}1 & S\ 0 & 1\end{pmatrix} \begin{pmatrix}\psi_{+}\ \psi_{-}\end{pmatrix}_{\text{old}}, $$
where the Stokes multiplier $S$ is typically $i$ or $-i$ for simple turning points.
References (selected)
- G. G. Stokes, “On the discontinuity of arbitrary constants which appear in divergent developments,” Trans. Cambridge Phil. Soc., 1886.
- F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
- C. J. Howls, “Hyperasymptotics for integrals with saddles,” Proceedings of the Royal Society A, 1992.
- M. V. Berry, “Uniform Asymptotic Smoothing of Stokes’s Discontinuities,” Proc. R. Soc. Lond. A, 1972.
These sources provide rigorous treatments of the Stokes phenomenon, its mathematical foundations, and its role in various areas of applied mathematics and theoretical physics.