---
tag: H5V4099GN7
kind: definition
name: Stokes phenomenon
msc: [34M40, 34M30, "34E05"]
defines: TTGF8KCA8C
uses: [AT1W4MMTYR, 0NAHXR1FRH, 0M3YQJ81NB]
sources:
  - url: https://en.wikipedia.org/wiki/Stokes_phenomenon
---

## Definition

A *sector* at $\infty$ is a set $S = \set{z \in \C : \abs{z} > R,\ \alpha < \arg z < \beta}$ with $R > 0$ and $\alpha < \beta$.
Let $f$ be [[holomorphic|0NAHXR1FRH]] on $S$, and let $m \ge 1$ be an integer.
Let $q$ be a polynomial in $z^{1/m}$, let $\mu \in \C$, and let $\sum_{k \ge 0} c_k z^{-k/m}$ be a formal series with $c_0 \ne 0$.
The function $f$ has the *asymptotic expansion* $f(z) \sim e^{q(z)} z^{\mu} \sum_{k \ge 0} c_k z^{-k/m}$ in $S$ if, for each $N \ge 0$ and each closed subsector $S'$ of $S$,
$$z^{N/m} \left( e^{-q(z)} z^{-\mu} f(z) - \sum_{k=0}^{N-1} c_k z^{-k/m} \right) \to c_N \quad \text{as } z \to \infty \text{ in } S'.$$
Let $f$ be a solution of a linear differential equation with an [[irregular singular point|AT1W4MMTYR]] at $\infty$, [[continued analytically|0M3YQJ81NB]] to a large sector.
The equation shows the *Stokes phenomenon* if some such solution has an asymptotic expansion of this form in a sector $S_1$ with the following property.
In an adjacent overlapping sector $S_2$, the expansion of the same solution is a different combination of such series.
So no single expansion of the solution holds on $S_1 \cup S_2$.

## Remarks

Consider the rays where a small term $e^{q_j(z)}$ is maximally subdominant compared with another term $e^{q_i(z)}$.
Its coefficient may jump there, and these rays are called Stokes lines in one common convention; other authors call them anti-Stokes lines.
Consider a system $Y' = A(z) Y$ where $A$ has a pole at $\infty$.
On two overlapping sectors, the actual solutions asymptotic to a fixed formal solution differ by a constant matrix, the *Stokes matrix*.
The phenomenon occurs exactly when some Stokes matrix is not the identity.
At a regular singular point it does not occur: the convergent Frobenius series hold in all directions.

Example: the Airy function satisfies $\operatorname{Ai}(z) \sim \frac{e^{-\zeta}}{2 \sqrt{\pi}\, z^{1/4}}$ with $\zeta = \frac{2}{3} z^{3/2}$ in the sector $\abs{\arg z} < \pi$.
On the negative real axis, however, $\operatorname{Ai}(-x) \sim \frac{1}{\sqrt{\pi}\, x^{1/4}} \sin\left( \frac{2}{3} x^{3/2} + \frac{\pi}{4} \right)$ for $x \to +\infty$, which is a sum of two exponential terms $e^{\pm i \frac{2}{3} x^{3/2}}$.

Non-example: the function $e^{z}$ solves $y' = y$, which has an irregular singular point at $\infty$, and $e^{z} \sim e^{z} \cdot 1$ holds in every sector; this solution shows no change of expansion.