---
tag: TTGF8KCA8C
kind: concept
name: Stokes phenomenon
msc: [34M40, 34M30, "34E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Stokes_phenomenon
same_as:
  wikidata: Q7618550
  wikipedia: Stokes phenomenon
---

## Summary

The Stokes phenomenon is the fact that one analytic function can have different asymptotic expansions in different sectors of the complex plane.
It occurs for solutions of linear differential equations at an [[irregular singular point|AT1W4MMTYR]].
Across certain rays, the Stokes lines, an exponentially small contribution in the expansion switches on or off.
The constants that measure the jump are the Stokes multipliers.
Together with the formal invariants, the Stokes data classify such equations up to meromorphic equivalence.
See the [[definition|H5V4099GN7]].

## Explanations

### L1 human

Approximate formulas for a function near infinity often hold only in some directions.
For the Airy function, one simple formula holds when $x$ is large and positive, where the function decays like an exponential.
For $x$ large and negative, the function oscillates, and the right formula is a sum of two exponentials, which together make a sine wave.
The function is the same everywhere, but its approximation changes form as one turns around the origin.
Stokes discovered this while computing the Airy integral in optics.