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  "body": "\n## Summary\n\nKummer's equation is the second order linear differential equation $z w'' + (b - z) w' - a w = 0$, with complex parameters $a$ and $b$.\nIt has a [[regular singular point|H96HXG6CA0]] at $0$ and an [[irregular singularity|AT1W4MMTYR]] at $\\infty$.\nIts solutions are the confluent hypergeometric functions, which include Laguerre polynomials, Hermite functions, the error function and the incomplete gamma function.\n\n## Explanations\n\n### L1 human\n\nKummer's equation arises when two of the three singular points of the hypergeometric equation are pushed together.\nOne replaces $z$ by $z / \\beta$ in the hypergeometric equation with parameters $a, \\beta, b$, and lets $\\beta \\to \\infty$; the singular points $\\beta$ and $\\infty$ merge into one.\nThe word \"confluent\" refers to this merging.\nA small example: for $a = b$, the function $w = e^z$ solves the equation, since $z e^z + (b - z) e^z - b e^z = 0$.\n",
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