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  "body": "\n## Statement\n\nLet $\\nu$ be a real number, and let $J_\\nu$ and $Y_\\nu$ be the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].\nLet $\\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $\\sim$ denote [[asymptotic equivalence|0QR92XBV6H]].\n\n1. (Large argument.) As $x \\to +\\infty$,\n$$J_\\nu(x) = \\sqrt{\\frac{2}{\\pi x}} \\left( \\cos\\left( x - \\frac{\\nu\\pi}{2} - \\frac{\\pi}{4} \\right) + O\\left( \\frac{1}{x} \\right) \\right),$$\n$$Y_\\nu(x) = \\sqrt{\\frac{2}{\\pi x}} \\left( \\sin\\left( x - \\frac{\\nu\\pi}{2} - \\frac{\\pi}{4} \\right) + O\\left( \\frac{1}{x} \\right) \\right).$$\n2. (Small argument.) As $x \\to 0^+$, if $\\nu$ is not a negative integer then $J_\\nu(x) \\sim \\frac{1}{\\Gamma(\\nu + 1)} \\left( \\frac{x}{2} \\right)^\\nu$; if $\\nu \u003e 0$ then $Y_\\nu(x) \\sim -\\frac{\\Gamma(\\nu)}{\\pi} \\left( \\frac{2}{x} \\right)^\\nu$; and $Y_0(x) \\sim \\frac{2}{\\pi} \\ln x$.\n\n## Remarks\n\nThe small-argument forms are the leading terms of the power series of $J_\\nu$ and of the series of $Y_\\nu$ obtained from its definition.\nThe large-argument forms come from an integral representation of the Hankel functions $H^{(1)}_\\nu = J_\\nu + iY_\\nu$ and the stationary-phase method.\nThey are the first terms of full asymptotic expansions in powers of $\\frac{1}{x}$.\nExample: for $\\nu = \\frac{1}{2}$, $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$ exactly, and indeed $\\cos\\left( x - \\frac{\\pi}{2} \\right) = \\sin x$, with no error term.\nThe large-argument form holds for fixed $\\nu$; it fails when $\\nu$ grows with $x$, for instance when $\\nu$ is close to $x$, where the transition is described by Airy functions.\n\n## Explanations\n\n### L1 human\n\nFar from the origin a Bessel function looks like a cosine wave whose amplitude shrinks like $\\frac{1}{\\sqrt{x}}$.\nThe order $\\nu$ only shifts the phase of the wave.\nNear the origin $J_\\nu$ behaves like the power $x^\\nu$, while $Y_\\nu$ blows up like $x^{-\\nu}$ or like $\\ln x$.\n\n### L2 human\n\nThe large-argument form explains why the positive zeros of $J_\\nu$ are eventually spaced by about $\\pi$.\nThe amplitude $\\frac{1}{\\sqrt{x}}$ reflects conservation of energy for cylindrical waves, whose wavefronts have length proportional to $x$.\nThe small-argument forms decide which solution is allowed in a boundary value problem on a disc: regularity at the centre excludes $Y_\\nu$.\nFor modified Bessel functions the analogues are $I_\\nu(x) \\sim \\frac{e^x}{\\sqrt{2\\pi x}}$ and $K_\\nu(x) \\sim \\sqrt{\\frac{\\pi}{2x}}\\, e^{-x}$.\n",
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