The abrupt change in the form of an asymptotic expansion of a function when the complex argument crosses certain rays (Stokes lines), manifested by the switching on or off of subdominant exponential contributions and governed by Stokes multipliers.

Principle

Analytic continuation of asymptotic series reveals hidden exponential terms whose contributions change discontinuously across special directions; correct asymptotics require uniform connection formulae that record these jumps.

Demonstration

The Airy function Ai(z) has asymptotic expansions that, as arg(z) crosses ±π/3, acquire or lose exponentially small terms; Stokes multipliers quantify the change between sectoral expansions.

Misapplication

Truncating a divergent asymptotic series in different complex sectors assuming the same remainder behavior ignores Stokes switching and leads to inconsistent analytic continuations.

Consequence

Accounting for the Stokes phenomenon yields consistent global approximations, predicts exponentially small corrections in certain sectors, and is essential in resurgent analysis and uniform asymptotics.

Reversal

If one ignores Stokes lines and treats asymptotic expansions as globally uniform, one misses sector-dependent exponentially small terms and obtains incorrect continuation across the complex plane.

Boundary

The phenomenon pertains to asymptotic expansions (often divergent) of analytic solutions to differential or integral equations in the complex plane; it does not describe ordinary convergence issues on the real axis absent sectoral analytic continuation.

Semantic Tension

Related but distinct from monodromy: monodromy concerns multi-valuedness under full circuit around singularities, while Stokes phenomenon concerns sudden changes in asymptotic content crossing rays even without topological winding.

Synthesis

The Stokes phenomenon is the sector-dependent switching of exponentially small terms in asymptotic expansions that occurs upon crossing Stokes lines, requiring connection data (Stokes multipliers) for global analytic description.