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  "body": "\n## Definition\n\nLet $a \\in \\C \\setminus \\set{0, 1}$ and let $q, \\alpha, \\beta, \\gamma, \\delta, \\epsilon$ be complex numbers with\n$$\\epsilon = \\alpha + \\beta + 1 - \\gamma - \\delta.$$\n*Heun's equation* is the [[linear differential equation|0QQRP17Z7S]]\n$$\\frac{d^2 w}{dz^2} + \\left( \\frac{\\gamma}{z} + \\frac{\\delta}{z - 1} + \\frac{\\epsilon}{z - a} \\right) \\frac{dw}{dz} + \\frac{\\alpha \\beta z - q}{z (z - 1)(z - a)}\\, w = 0.$$\nAssume $\\gamma \\notin \\set{0, -1, -2, \\dots}$.\nThe *local Heun function* $Hl(a, q; \\alpha, \\beta, \\gamma, \\delta; z)$ is the unique solution of Heun's equation that is analytic at $z = 0$ and takes the value $1$ there.\n\n## Remarks\n\nHeun's equation is a [[Fuchsian equation|ZBVH2YE50B]] with [[regular singular points|H96HXG6CA0]] $0, 1, a, \\infty$.\nEvery Fuchsian equation with four regular singular points takes this form after moving the singular points there and making one exponent $0$ at each finite point.\nThe exponents are $\\set{0, 1 - \\gamma}$ at $0$, $\\set{0, 1 - \\delta}$ at $1$, $\\set{0, 1 - \\epsilon}$ at $a$ and $\\set{\\alpha, \\beta}$ at $\\infty$.\nThe relation $\\epsilon = \\alpha + \\beta + 1 - \\gamma - \\delta$ is Fuchs' relation: the sum of all exponents equals $2$.\nThe number $q$ is the accessory parameter; it is not determined by the exponents.\nThe power series of $Hl$ at $0$ converges in the disc $\\abs{z} \u003c \\min(1, \\abs{a})$.\nExample: for $q = 0$ and $\\alpha = 0$, the constant $w = 1$ is a solution, since $w' = w'' = 0$ and $\\alpha\\beta z - q = 0$; so $Hl(a, 0; 0, \\beta, \\gamma, \\delta; z) = 1$.\nNon-example: for $q = 1$ and $\\alpha = 0$, the constant $1$ is not a solution, since the equation leaves the term $\\frac{-1}{z (z - 1)(z - a)} \\ne 0$.\nA *Heun function* in the narrow sense is a local solution that is analytic at two singular points at once, for instance at $0$ and $1$.\nThis happens only for special values of $q$.\n",
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