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  "body": "\n## Summary\n\nA pseudoanalytic function is a complex function that is \"holomorphic relative to a pair of basis functions\" $(F, G)$ instead of relative to the pair $(1, i)$.\nThe notion was introduced by Lipman Bers to extend the theory of [[holomorphic functions|0NAHXR1FRH]] to solutions of first-order elliptic systems in the plane.\nFor smooth generating pairs, pseudoanalytic functions satisfy equations of the form $w_{\\bar z} = a w + b \\bar w$, and many theorems of complex analysis carry over to them.\n\n## Explanations\n\n### L1 human\n\nEvery complex number $w$ can be written as $\\varphi \\cdot 1 + \\psi \\cdot i$ with $\\varphi, \\psi$ real, and $w$ is holomorphic when these real coefficients vary in a special way.\nA pseudoanalytic function replaces the fixed pair $(1, i)$ by a pair of functions $(F, G)$ that are independent over the reals at each point.\nOne writes $w = \\varphi F + \\psi G$ and asks the real coefficients to satisfy the analogue of the Cauchy–Riemann condition.\nWith $F = 1$ and $G = i$ this is exactly ordinary holomorphy.\n",
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