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  "body": "\n## Definition\n\nLet $\\nu$ be real and let $x \u003e 0$.\nLet $J_\\nu$ be the [[Bessel function of the first kind|CZCMRDWDYW]] and $K_\\nu$ the [[modified Bessel function|2ET068BZZH]] of the second kind, with principal branches.\nThe *Kelvin functions* of order $\\nu$ are the real functions $\\operatorname{ber}_\\nu$, $\\operatorname{bei}_\\nu$, $\\operatorname{ker}_\\nu$, $\\operatorname{kei}_\\nu$ on $(0, \\infty)$ defined by\n$$\\operatorname{ber}_\\nu(x) + i \\operatorname{bei}_\\nu(x) = J_\\nu\\left( x e^{3\\pi i/4} \\right),$$\n$$\\operatorname{ker}_\\nu(x) + i \\operatorname{kei}_\\nu(x) = e^{-\\nu\\pi i/2}\\, K_\\nu\\left( x e^{\\pi i/4} \\right).$$\n\n## Remarks\n\nThe functions $J_\\nu(x e^{3\\pi i/4})$ and $K_\\nu(x e^{\\pi i/4})$ solve $x^2 w'' + x w' - (i x^2 + \\nu^2) w = 0$, since $\\left( x e^{3\\pi i/4} \\right)^2 = -i x^2$.\nExample: for $\\nu = 0$, $J_0(z) = \\sum_m \\frac{(-1)^m (z/2)^{2m}}{(m!)^2}$ and $(-1)^m e^{3\\pi i m/2} = i^m$, so the real part keeps the terms with $m = 2k$ and gives $\\operatorname{ber}_0(x) = \\sum_{k \\ge 0} \\frac{(-1)^k (x/2)^{4k}}{((2k)!)^2}$.\nThe odd terms give $\\operatorname{bei}_0(x) = \\sum_{k \\ge 0} \\frac{(-1)^k (x/2)^{4k+2}}{((2k+1)!)^2}$.\nA non-example: $J_0(x)$ on the real axis is not $\\operatorname{ber}_0(x)$; its series has the term $-\\frac{x^2}{4}$, which $\\operatorname{ber}_0$ lacks, because the argument is not rotated to the ray $\\arg z = \\frac{3\\pi}{4}$.\nWhen $\\nu = 0$ the index is usually dropped: $\\operatorname{ber} = \\operatorname{ber}_0$, and so on.\n",
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