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  "body": "\n## Summary\n\nHeun functions are solutions of Heun's equation, the general second order linear [[Fuchsian equation|ZBVH2YE50B]] with four [[regular singular points|H96HXG6CA0]], placed at $0, 1, a$ and $\\infty$.\nThe local Heun function is the solution that is analytic at $0$ with value $1$.\nHeun's equation is the next case after the [[hypergeometric differential equation|W8VE25R5A7]], which has three regular singular points.\nIt contains the Lamé, Mathieu and spheroidal wave equations as special or confluent cases.\n\n## Explanations\n\n### L1 human\n\nThe hypergeometric equation is fixed, up to its parameters, by the positions of its three singular points and the exponents there.\nWith four singular points this is no longer true: besides the exponents there is one extra number $q$, the accessory parameter.\nThis extra freedom makes Heun functions much harder to handle than hypergeometric functions.\nThey come up in quantum mechanics and in general relativity, among other places.\nA trivial example: when $q = 0$ and $\\alpha = 0$, the constant function $1$ solves Heun's equation.\n",
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