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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^4$ be open and let $C$ be a piecewise smooth closed curve in $\\C$.\nFor $x = (x_1, x_2, x_3, x_4) \\in U$ and $\\zeta \\in \\C$ put\n$$A(x, \\zeta) = (x_1 + i x_2) + (x_4 + i x_3)\\, \\zeta, \\qquad B(x, \\zeta) = (x_4 - i x_3) - (x_1 - i x_2)\\, \\zeta.$$\nLet $\\Omega \\subseteq \\C^3$ be an open set that contains the points $(A(x, \\zeta), B(x, \\zeta), \\zeta)$ for all $x \\in U$ and $\\zeta \\in C$.\nLet $f(w_1, w_2, \\zeta)$ be a [[holomorphic function|0NAHXR1FRH]] on $\\Omega$.\nThe *Bateman transform* of $f$ is the function $u \\colon U \\to \\C$ given by\n$$u(x) = \\oint_C f\\big(A(x, \\zeta), B(x, \\zeta), \\zeta\\big)\\, d\\zeta.$$\n\n## Remarks\n\nThe function $u$ is [[harmonic|0DEB0JMN0W]] on $U$, that is, it solves [[Laplace's equation|S8RG6N1PRT]] $\\sum_{k=1}^4 \\partial_k^2 u = 0$.\nIndeed, by the chain rule, $\\sum_k \\partial_k^2$ applied to $f(A, B, \\zeta)$ is a combination of second derivatives of $f$ with coefficients $\\sum_k (\\partial_k A)^2$, $\\sum_k \\partial_k A\\, \\partial_k B$ and $\\sum_k (\\partial_k B)^2$.\nThe gradients are $\\nabla A = (1, i, i\\zeta, \\zeta)$ and $\\nabla B = (-\\zeta, i\\zeta, -i, 1)$, and all three sums vanish: $1 - 1 - \\zeta^2 + \\zeta^2 = 0$, $-\\zeta - \\zeta + \\zeta + \\zeta = 0$ and $\\zeta^2 - \\zeta^2 - 1 + 1 = 0$.\nDifferentiation under the integral sign then gives $\\Delta u = 0$.\nWhen $\\Omega$ also contains the points $(A(x, \\zeta), B(x, \\zeta), \\zeta)$ with $x_4$ replaced by $i t$, the same integral defines $u(x_1, x_2, x_3, i t)$.\nReplacing $x_4$ by $i t$ turns $\\partial_{x_4}^2$ into $-\\partial_t^2$, so $u(x_1, x_2, x_3, it)$ solves the [[wave equation|5EHD2XBQDG]] $\\partial_t^2 u = \\partial_1^2 u + \\partial_2^2 u + \\partial_3^2 u$.\n\nExample: $f(w_1, w_2, \\zeta) = \\frac{w_1}{\\zeta^2}$ and $C$ the unit circle.\nThen $u(x) = \\oint_C \\Big( \\frac{x_1 + i x_2}{\\zeta^2} + \\frac{x_4 + i x_3}{\\zeta} \\Big) d\\zeta = 2\\pi i\\, (x_4 + i x_3)$, a linear, hence harmonic, function.\n\nNon-example: if $f(w_1, w_2, \\zeta) = \\overline{w_1}$, which is not holomorphic, the function $\\oint_C \\overline{A(x, \\zeta)}\\, d\\zeta$ is not covered by the definition, and the chain rule computation above fails.\n",
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