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  "body": "\n## Definition\n\nLet $a$ and $b$ be complex numbers.\n*Kummer's equation* with parameters $a$ and $b$ is the [[linear differential equation|0QQRP17Z7S]] of second order\n$$z \\frac{d^2 w}{dz^2} + (b - z) \\frac{dw}{dz} - a w = 0$$\nfor a function $w$ of a complex variable $z$.\n\n## Remarks\n\nThe point $z = 0$ is a [[regular singular point|H96HXG6CA0]] with exponents $0$ and $1 - b$.\nThe point $z = \\infty$ is an [[irregular singularity|AT1W4MMTYR]], and there are no other singular points.\nExample: for $a = b$, the function $w = e^z$ is a solution, since $z e^z + (b - z) e^z - b e^z = 0$.\nNon-example: for $a \\ne b$, the same function is not a solution, since the left side equals $(b - a) e^z \\ne 0$.\nFor $b$ not in $\\set{0, -1, -2, \\dots}$, the solution analytic at $0$ with value $1$ is Kummer's function $M(a, b, z) = {}_1F_1(a; b; z)$.\nA second standard solution is the Tricomi function $U(a, b, z)$, singled out by its behaviour $U(a, b, z) \\sim z^{-a}$ as $z \\to \\infty$ in the right half-plane.\nThe substitution $w = z^{-b/2} e^{z/2} v$ turns the equation into Whittaker's equation, which has no first derivative term.\n",
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