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  "body": "\n## Summary\n\nResurgent functions are a class of analytic functions introduced by Jean Écalle to study divergent series.\nA resurgent series is a formal series whose Borel transform converges near $0$.\nThis Borel transform must also continue analytically along every path that avoids a discrete set of singular points.\nMany divergent series of analysis are resurgent, such as [[Stirling's series|K29QYWB6A5]] and the formal solutions of many analytic differential equations.\nThe singularities in the Borel plane link the divergent series to exponentially small terms.\nThis is the base of [[Borel summation|8DJPSYSHGJ]] beyond its simplest cases.\n\n## Explanations\n\n### L1 human\n\nA divergent series with factorially growing coefficients can be tamed by dividing its $n$-th coefficient by $n!$.\nThe result is a convergent power series, a function on a small disc.\nThe series is resurgent when this function can be continued everywhere, except at some isolated points that can be walked around.\nFor example, $\\sum_{n \\ge 0} (-1)^n n! \\, z^{-n-1}$ becomes $\\sum_{n \\ge 0} (-1)^n \\zeta^n = \\frac{1}{1 + \\zeta}$, which continues to all of $\\C$ except $\\zeta = -1$.\nThe name says that the same information \"resurges\" at each singular point.\n",
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