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  "body": "\n## Definition\n\nLet $\\nu \u003e -1$ and let $J_\\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\\nu$, a solution of [[Bessel's equation|EAV97QH9DF]].\nLet $0 \u003c j_{\\nu,1} \u003c j_{\\nu,2} \u003c \\cdots$ be the positive zeros of $J_\\nu$.\nLet $f$ be a measurable function on $(0, 1)$ with $\\int_0^1 x \\abs{f(x)}^2 \\, dx \u003c \\infty$.\nThe *Fourier–Bessel series* of $f$ of order $\\nu$ is\n$$\\sum_{k=1}^{\\infty} c_k \\, J_\\nu(j_{\\nu,k} x), \\qquad c_k = \\frac{2}{J_{\\nu+1}(j_{\\nu,k})^2} \\int_0^1 x \\, f(x) \\, J_\\nu(j_{\\nu,k} x) \\, dx.$$\n\n## Remarks\n\nThe functions $J_\\nu(j_{\\nu,k} x)$ are orthogonal in the weighted [[Lebesgue space|SASP14MA1Z]] $L^2((0, 1), x \\, dx)$, with\n$$\\int_0^1 x \\, J_\\nu(j_{\\nu,k} x) \\, J_\\nu(j_{\\nu,m} x) \\, dx = \\frac{\\delta_{km}}{2} J_{\\nu+1}(j_{\\nu,k})^2.$$\nSo the series is the [[generalized Fourier series|SNJC87TBHD]] of $f$ with respect to this orthogonal family, and the formula for $c_k$ is the usual quotient of inner products.\nThe family is complete in $L^2((0, 1), x \\, dx)$, so the series converges to $f$ in this space.\nOn an interval $[0, b]$ one uses $J_\\nu(j_{\\nu,k} x / b)$ instead.\n\nExample: $\\nu = 0$ and $f = 1$.\nSince $\\frac{d}{dt}\\big(t J_1(t)\\big) = t J_0(t)$, the substitution $t = j x$ gives $\\int_0^1 x J_0(jx) \\, dx = J_1(j)/j$.\nHence $c_k = \\frac{2}{j_{0,k} J_1(j_{0,k})}$ and\n$$1 = \\sum_{k=1}^{\\infty} \\frac{2}{j_{0,k} \\, J_1(j_{0,k})} \\, J_0(j_{0,k} x) \\quad \\text{in } L^2((0,1), x\\,dx).$$\n\nNon-example: the functions $J_0(k\\pi x)$, $k = 1, 2, \\dots$, do not give a Fourier–Bessel series.\nThe numbers $k\\pi$ are not zeros of $J_0$; for instance $J_0(\\pi) \\approx -0.304 \\ne 0$.\n",
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