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  "body": "\n## Definition\n\nLet $\\nu \\in \\C$, and let $J_\\nu$ and $Y_\\nu$ be the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]] on $\\C \\setminus (-\\infty, 0]$.\nThe *Hankel functions* of order $\\nu$ are\n$$H^{(1)}_\\nu(z) = J_\\nu(z) + i\\, Y_\\nu(z), \\qquad H^{(2)}_\\nu(z) = J_\\nu(z) - i\\, Y_\\nu(z).$$\n\n## Remarks\n\nBoth functions solve Bessel's equation $z^2 y'' + z y' + (z^2 - \\nu^2) y = 0$, since they are linear combinations of $J_\\nu$ and $Y_\\nu$.\nThey are independent: their Wronskian is $W(H^{(1)}_\\nu, H^{(2)}_\\nu)(z) = -\\frac{4i}{\\pi z}$.\nExample: for $\\nu = \\frac{1}{2}$ and $x \u003e 0$, $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$ and $Y_{1/2}(x) = -\\sqrt{\\frac{2}{\\pi x}} \\cos x$, so $H^{(1)}_{1/2}(x) = -i \\sqrt{\\frac{2}{\\pi x}}\\, e^{ix}$.\nFor real $\\nu$ and $x \u003e 0$, $H^{(2)}_\\nu(x)$ is the complex conjugate of $H^{(1)}_\\nu(x)$.\nA non-example: $J_\\nu + Y_\\nu$ also solves Bessel's equation, but it is not a Hankel function, because the coefficient of $Y_\\nu$ is not $\\pm i$.\nThey are also called Bessel functions of the third kind.\n",
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