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  "body": "\n## Summary\n\nAn irregular singularity of a linear differential equation is a [[singular point|QK2J038VB4]] that is not a [[regular singular point|H96HXG6CA0]].\nNear it, solutions can grow or decay faster than any power, typically like $e^{c / (z - z_0)^k}$ or $e^{c z^k}$ near $\\infty$.\nThe Frobenius method fails there, and the solutions are described by asymptotic expansions valid only in sectors.\nThis leads to the [[Stokes phenomenon|TTGF8KCA8C]].\nSee the [[definition|A6EXFAP5EQ]].\n\n## Explanations\n\n### L1 human\n\nAt a regular singular point solutions behave like powers of the distance to the point.\nAt an irregular one they can behave like exponentials of a negative power, which is much wilder.\nFor $y' = \\frac{y}{z^2}$, the solution is $y = e^{-1/z}$, which has an essential singularity at $0$.\nThe Bessel and Airy equations have an irregular singular point at $\\infty$, where their solutions oscillate or grow exponentially.\n",
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  "name": "Irregular singularity",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/AT1/AT1W4MMTYR.md",
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  "tag": "AT1W4MMTYR",
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  "url": "https://math2.tamnd.com/wiki/irregular-singularity/",
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