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  "body": "\n## Summary\n\nThe Bateman transform builds solutions of [[Laplace's equation|S8RG6N1PRT]] in four real variables from holomorphic functions of three complex variables.\nIt also gives solutions of the [[wave equation|5EHD2XBQDG]] in three space dimensions.\nThe solution is a contour integral in one complex variable $\\zeta$ of a holomorphic function evaluated at two linear expressions in the coordinates.\nIt is named after Harry Bateman and extends Whittaker's integral formula for harmonic functions in three variables.\nIt is a forerunner of the Penrose transform of twistor theory.\n\n## Explanations\n\n### L1 human\n\nHolomorphic functions of one variable give harmonic functions of two variables for free: their real parts.\nThe Bateman transform does something similar one level up.\nIt plugs carefully chosen combinations of four real coordinates into a holomorphic function and integrates along a loop.\nThe combinations are chosen so that the Laplacian of the result vanishes automatically, whatever the holomorphic function is.\n",
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