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  "body": "\n## Definition\n\nLet $\\nu$ be a complex number and let $\\Gamma$ be the [[gamma function|0H5SYBSED9]], with $\\frac{1}{\\Gamma(w)}$ read as $0$ when $w \\in \\set{0, -1, -2, \\dots}$.\nFor $z \\in \\C \\setminus (-\\infty, 0]$, with the principal branch of $\\left( \\frac z2 \\right)^\\nu$, the *Bessel function of the first kind* of order $\\nu$ is\n$$J_\\nu(z) = \\left( \\frac z2 \\right)^{\\nu} \\sum_{k = 0}^{\\infty} \\frac{(-1)^k}{k!\\, \\Gamma(k + \\nu + 1)} \\left( \\frac z2 \\right)^{2k}.$$\nIf $\\nu = n$ is an integer, the same formula defines $J_n$ as an entire function of $z$.\n\n## Remarks\n\nThe [[power series|0MZB06ST2B]] in $\\left( \\frac z2 \\right)^2$ converges for all $z$, since the ratio of consecutive terms is $\\frac{-(z/2)^2}{(k + 1)(k + \\nu + 1)} \\to 0$.\nThe function $J_\\nu$ solves [[Bessel's equation|EAV97QH9DF]] $z^2 y'' + z y' + (z^2 - \\nu^2) y = 0$.\nExample: $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$ for $x \u003e 0$.\nCheck: with $\\Gamma(k + \\frac32) = \\frac{(2k + 1)!\\, \\sqrt{\\pi}}{4^k\\, k!\\, 2}$, the term $k$ equals $\\left( \\frac x2 \\right)^{1/2} \\frac{2 (-1)^k x^{2k}}{\\sqrt{\\pi}\\, (2k + 1)!}$, and summing gives $\\sqrt{\\frac{2}{\\pi x}} \\sum_k \\frac{(-1)^k x^{2k + 1}}{(2k + 1)!}$.\nFor a positive integer $n$ and $\\nu = -n$, the terms with $k \u003c n$ vanish, and one gets $J_{-n} = (-1)^n J_n$.\nSo $J_n$ and $J_{-n}$ are not independent, and a second solution is the Bessel function of the second kind.\nFor integer $n$, the generating function is $\\exp\\left( \\frac z2 \\left( t - \\frac1t \\right) \\right) = \\sum_{n \\in \\Z} J_n(z)\\, t^n$ for $t \\ne 0$.\nNon-example: $\\cos z$ is not $J_0(z)$, although both equal $1$ at $0$; their series differ already in the $z^2$ term, $-\\frac{z^2}{2}$ for $\\cos z$ against $-\\frac{z^2}{4}$ for $J_0(z)$.\n",
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