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  "body": "\n## Summary\n\nThe Hankel functions $H^{(1)}_\\nu$ and $H^{(2)}_\\nu$ are the Bessel functions of the third kind.\nThey are the complex combinations $J_\\nu \\pm i Y_\\nu$ of the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].\nThey play the role of $e^{\\pm i x}$ next to the roles of $\\cos x$ and $\\sin x$ played by $J_\\nu$ and $Y_\\nu$.\nThey describe outgoing and incoming cylindrical waves.\n\n## Explanations\n\n### L1 human\n\nFor the equation $y'' + y = 0$, one can use the real solutions $\\cos x$ and $\\sin x$ or the complex solutions $e^{ix}$ and $e^{-ix}$.\nBessel's equation has the same choice.\nThe pair $J_\\nu$, $Y_\\nu$ is the real one, and the Hankel functions are the complex one.\nFor large $x \u003e 0$, $H^{(1)}_\\nu(x)$ is close to $\\sqrt{\\frac{2}{\\pi x}}\\, e^{i(x - \\nu\\pi/2 - \\pi/4)}$, a wave travelling outward whose size shrinks.\n",
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