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  "body": "\n## Definition\n\nLet $\\nu \\in \\C$ and $z \\in \\C$.\nThe *Anger function* of order $\\nu$ is\n$$\\mathbf{J}_\\nu(z) = \\frac{1}{\\pi} \\int_0^\\pi \\cos(\\nu\\tau - z \\sin\\tau)\\, d\\tau.$$\nThe *Weber function* of order $\\nu$ is\n$$\\mathbf{E}_\\nu(z) = \\frac{1}{\\pi} \\int_0^\\pi \\sin(\\nu\\tau - z \\sin\\tau)\\, d\\tau.$$\n\n## Remarks\n\nBoth are entire functions of $z$ and of $\\nu$, since the integrands are entire and the interval is compact.\nIf $n$ is an integer, then $\\mathbf{J}_n = J_n$, the [[Bessel function of the first kind|CZCMRDWDYW]], by Bessel's integral.\nExample: at $z = 0$, $\\mathbf{J}_\\nu(0) = \\frac{1}{\\pi} \\int_0^\\pi \\cos(\\nu\\tau)\\, d\\tau = \\frac{\\sin(\\nu\\pi)}{\\nu\\pi}$ for $\\nu \\ne 0$.\nA non-example: for $\\nu = \\frac{1}{2}$, $\\mathbf{J}_{1/2}(0) = \\frac{2}{\\pi} \\ne 0$, while $J_{1/2}(0) = 0$, so $\\mathbf{J}_{1/2}$ is not the Bessel function $J_{1/2}$.\nThe Anger function solves $z^2 w'' + z w' + (z^2 - \\nu^2)\\, w = \\frac{(z - \\nu)\\sin(\\nu\\pi)}{\\pi}$, so it solves Bessel's equation exactly when $\\nu$ is an integer.\nThe Weber function $\\mathbf{E}_\\nu$ here is different from the Bessel function of the second kind, which is also sometimes called a Weber function.\n",
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