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  "body": "\n## Statement\n\nLet $J_n$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of integer order $n$.\nFor all $z \\in \\C$ and $\\theta \\in \\R$,\n$$e^{iz \\cos\\theta} = \\sum_{n=-\\infty}^{\\infty} i^n J_n(z)\\, e^{in\\theta}, \\qquad e^{iz \\sin\\theta} = \\sum_{n=-\\infty}^{\\infty} J_n(z)\\, e^{in\\theta},$$\nand both series converge absolutely and uniformly in $\\theta$.\n\n## Remarks\n\nThe generating function of the Bessel functions is $\\exp\\left( \\frac{z}{2} \\left( t - \\frac{1}{t} \\right) \\right) = \\sum_{n \\in \\Z} J_n(z)\\, t^n$ for $t \\ne 0$, obtained by multiplying the series of $e^{zt/2}$ and $e^{-z/(2t)}$.\nPutting $t = e^{i\\theta}$ gives the second formula, since $\\frac{1}{2}(t - t^{-1}) = i\\sin\\theta$.\nReplacing $\\theta$ by $\\theta + \\frac{\\pi}{2}$ turns $\\sin\\theta$ into $\\cos\\theta$ and gives the first formula.\nSo the expansion is the [[Fourier series|G12KXDHPPS]] of the periodic function $\\theta \\mapsto e^{iz\\sin\\theta}$, and its Fourier coefficients are $J_n(z)$.\nExample: at $\\theta = 0$ the second formula gives $1 = \\sum_{n} J_n(z) = J_0(z) + 2\\sum_{k \\ge 1} J_{2k}(z)$, using $J_{-n} = (-1)^n J_n$.\n\n## Explanations\n\n### L1 human\n\nA plane wave $e^{iz\\cos\\theta}$ can be broken up into circular pieces, one for each whole number $n$.\nThe size of the $n$-th piece is the Bessel function $J_n(z)$.\nThis is why Bessel functions appear whenever a straight wave meets something round.\n\n### L2 human\n\nTaking real and imaginary parts gives expansions of $\\cos(z\\sin\\theta)$ and $\\sin(z\\sin\\theta)$ used in frequency modulation, where the strengths of the sidebands are Bessel values.\nReading off a Fourier coefficient gives Bessel's integral for $J_n$.\nIn two-dimensional scattering it expands an incident plane wave in cylindrical waves, the first step of the partial-wave method.\nMultiplying two such expansions gives addition theorems for Bessel functions.\n",
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  "msc": [
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  "name": "Jacobi–Anger expansion",
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