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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer and let $x \u003e 0$.\nLet $J_\\nu$ and $Y_\\nu$ be the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].\nThe *spherical Bessel functions* of order $n$ are\n$$j_n(x) = \\sqrt{\\frac{\\pi}{2x}}\\, J_{n + 1/2}(x), \\qquad y_n(x) = \\sqrt{\\frac{\\pi}{2x}}\\, Y_{n + 1/2}(x).$$\n\n## Remarks\n\nThey solve $x^2 y'' + 2x y' + \\left( x^2 - n(n+1) \\right) y = 0$, which is the radial equation of $\\Delta u + u = 0$ in spherical coordinates.\nExample: $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$, so $j_0(x) = \\frac{\\sin x}{x}$; one checks $x^2 j_0'' + 2x j_0' + x^2 j_0 = x (x j_0)'' + x^2 j_0 = x (\\sin x)'' + x \\sin x = 0$.\nLikewise $y_0(x) = -\\frac{\\cos x}{x}$.\nIn general $j_n(x) = (-x)^n \\left( \\frac{1}{x} \\frac{d}{dx} \\right)^n \\frac{\\sin x}{x}$.\nThe spherical Hankel functions are $h^{(1)}_n = j_n + i y_n$ and $h^{(2)}_n = j_n - i y_n$; for instance $h^{(1)}_0(x) = -i \\frac{e^{ix}}{x}$.\nA non-example: $J_n(x)$ for an integer $n$ is a cylindrical, not a spherical, Bessel function; it is not elementary.\n",
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