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  "body": "\n## Summary\n\nThe Kelvin functions $\\operatorname{ber}_\\nu$, $\\operatorname{bei}_\\nu$, $\\operatorname{ker}_\\nu$ and $\\operatorname{kei}_\\nu$ are the real and imaginary parts of Bessel functions evaluated along the ray $\\arg z = \\frac{3\\pi}{4}$ or $\\frac{\\pi}{4}$.\nThey are real functions of a positive real variable.\nThey describe alternating current in a round wire, where the current crowds toward the surface (the skin effect).\n\n## Explanations\n\n### L1 human\n\nSome physical problems lead to Bessel's equation with a complex coefficient, such as $x^2 w'' + x w' - i x^2 w = 0$.\nIts solutions are complex numbers, and one splits them into real and imaginary parts.\nThese parts have names: $\\operatorname{ber}$ and $\\operatorname{bei}$ for the solution that is finite at $0$, and $\\operatorname{ker}$ and $\\operatorname{kei}$ for the one that decays at infinity.\nFor example, $\\operatorname{ber}_0(x) = 1 - \\frac{(x/2)^4}{(2!)^2} + \\frac{(x/2)^8}{(4!)^2} - \\cdots$.\n",
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