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  "body": "\n## Definition\n\nA *sector* at $\\infty$ is a set $S = \\set{z \\in \\C : \\abs{z} \u003e R,\\ \\alpha \u003c \\arg z \u003c \\beta}$ with $R \u003e 0$ and $\\alpha \u003c \\beta$.\nLet $f$ be [[holomorphic|0NAHXR1FRH]] on $S$, and let $m \\ge 1$ be an integer.\nLet $q$ 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 $c_0 \\ne 0$.\nThe 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 $S$,\n$$z^{N/m} \\left( e^{-q(z)} z^{-\\mu} f(z) - \\sum_{k=0}^{N-1} c_k z^{-k/m} \\right) \\to c_N \\quad \\text{as } z \\to \\infty \\text{ in } S'.$$\nLet $f$ be a solution of a linear differential equation with an [[irregular singular point|AT1W4MMTYR]] at $\\infty$, [[continued analytically|0M3YQJ81NB]] to a large sector.\nThe 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.\nIn an adjacent overlapping sector $S_2$, the expansion of the same solution is a different combination of such series.\nSo no single expansion of the solution holds on $S_1 \\cup S_2$.\n\n## Remarks\n\nConsider the rays where a small term $e^{q_j(z)}$ is maximally subdominant compared with another term $e^{q_i(z)}$.\nIts coefficient may jump there, and these rays are called Stokes lines in one common convention; other authors call them anti-Stokes lines.\nConsider a system $Y' = A(z) Y$ where $A$ has a pole at $\\infty$.\nOn two overlapping sectors, the actual solutions asymptotic to a fixed formal solution differ by a constant matrix, the *Stokes matrix*.\nThe phenomenon occurs exactly when some Stokes matrix is not the identity.\nAt a regular singular point it does not occur: the convergent Frobenius series hold in all directions.\n\nExample: 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 $\\abs{\\arg z} \u003c \\pi$.\nOn 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 $e^{\\pm i \\frac{2}{3} x^{3/2}}$.\n\nNon-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.\n",
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