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  "body": "\n## Example\n\nLet $\\nu \\in \\C$. *Bessel's equation* of order $\\nu$ is\n$$z^2 y'' + z y' + (z^2 - \\nu^2)\\, y = 0.$$\nIn normalised form, $p(z) = \\frac{1}{z}$ and $q(z) = 1 - \\frac{\\nu^2}{z^2}$.\nThen $z p(z) = 1$ and $z^2 q(z) = z^2 - \\nu^2$ are analytic at $0$, so $0$ is a [[regular singular point|H96HXG6CA0]].\nThe indicial equation is $r (r - 1) + r - \\nu^2 = r^2 - \\nu^2 = 0$, with exponents $r = \\pm \\nu$.\nIf $\\nu$ is not a negative integer, the [[Frobenius method|W8ZWXC2HQS]] with $r = \\nu$ gives the recurrence $\\left( (\\nu + k)^2 - \\nu^2 \\right) a_k = - a_{k-2}$, that is, $k (k + 2 \\nu)\\, a_k = - a_{k-2}$, with $a_1 = 0$.\nWith $a_0 = \\frac{1}{2^\\nu \\Gamma(\\nu + 1)}$ it gives the Bessel function of the first kind\n$$J_\\nu(z) = \\sum_{m \\ge 0} \\frac{(-1)^m}{m!\\, \\Gamma(m + \\nu + 1)} \\left( \\frac{z}{2} \\right)^{2m + \\nu},$$\nwhere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nThe series converges for all $z$, since the coefficients of the equation are analytic on $\\C \\setminus \\set{0}$.\nAt $\\infty$, the function $z^2 q(z) = z^2 - \\nu^2$ is not bounded, so $\\infty$ is an irregular singular point.\n\n## Remarks\n\nIf $\\nu \\notin \\Z$, then $J_\\nu$ and $J_{-\\nu}$ are linearly independent solutions.\nIf $\\nu$ is an integer, then $J_{-\\nu} = (-1)^\\nu J_\\nu$, and a second solution contains a logarithm, as Fuchs' theorem predicts.\nFor $\\nu = \\frac{1}{2}$, one gets $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}}\\, \\sin x$ for $x \u003e 0$.\nThe equation arises from separating variables in the Laplace or wave equation in cylindrical coordinates.\n\n## Explanations\n\n### L1 human\n\nBessel's equation describes waves on a round drum and heat flow in a cylinder.\nIts solutions near $0$ behave like the power $z^{\\nu}$, and the Frobenius method finds them as series.\nFar from $0$ they oscillate like a sine wave whose size decays like $\\frac{1}{\\sqrt{x}}$.\n",
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