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  "body": "\n## Summary\n\nThe Bessel function of the first kind $J_\\nu(z)$ is the solution of [[Bessel's equation|EAV97QH9DF]] $z^2 y'' + z y' + (z^2 - \\nu^2) y = 0$ that behaves like $\\frac{1}{\\Gamma(\\nu + 1)} \\left( \\frac z2 \\right)^\\nu$ near $z = 0$.\nThis description assumes that $\\nu$ is not a negative integer.\nIt is given by a power series that converges for every $z$.\nBessel functions describe waves and heat flow in cylinders and discs, such as the vibration modes of a drum.\nThey appear throughout analysis, number theory and probability.\n\n## Explanations\n\n### L1 human\n\nA Bessel function of the first kind looks like a cosine wave whose height slowly shrinks, roughly like $1 / \\sqrt{x}$.\nFor example $J_0(x)$ starts at $1$ for $x = 0$ and then oscillates around $0$ with decreasing amplitude.\nIts zeros give the frequencies at which a circular drum can vibrate.\nFor half-integer order the function is elementary: $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$ for $x \u003e 0$.\n",
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