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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\C$ be a [[domain|0EWMDD6C14]].\nA *generating pair* on $\\Omega$ is a pair $(F, G)$ of [[Hölder continuous|0WZTM0EFR6]] functions $\\Omega \\to \\C$ with $\\operatorname{Im}\\left( \\overline{F(z)}\\, G(z) \\right) \u003e 0$ for all $z \\in \\Omega$.\nThen every function $w \\colon \\Omega \\to \\C$ can be written uniquely as $w = \\varphi F + \\psi G$ with $\\varphi, \\psi \\colon \\Omega \\to \\R$.\nThe function $w$ is *$(F, G)$-pseudoanalytic* if $\\varphi$ and $\\psi$ have continuous first partial derivatives and\n$$\\varphi_{\\bar z}\\, F + \\psi_{\\bar z}\\, G = 0 \\quad \\text{on } \\Omega,$$\nwhere $\\partial_{\\bar z} = \\frac{1}{2}\\left( \\partial_x + i \\partial_y \\right)$ is the [[Wirtinger derivative|WMXDVF36ZT]].\n\n## Remarks\n\nThe condition $\\operatorname{Im}(\\bar F G) \u003e 0$ says that $F(z)$ and $G(z)$ are linearly independent over $\\R$, which gives the uniqueness of $\\varphi$ and $\\psi$.\n\nExample: for $(F, G) = (1, i)$, the condition reads $(\\varphi + i \\psi)_{\\bar z} = 0$.\nSo the $(1, i)$-pseudoanalytic functions are exactly the [[holomorphic functions|0NAHXR1FRH]] with continuous derivative.\n\nNon-example: $w(z) = \\bar z$ is not $(1, i)$-pseudoanalytic.\nHere $\\varphi = x$ and $\\psi = -y$, so $\\varphi_{\\bar z} + i \\psi_{\\bar z} = \\frac{1}{2} + i \\cdot \\left( -\\frac{i}{2} \\right) = 1 \\ne 0$.\n\nIf $F$ and $G$ are continuously differentiable, then $w_{\\bar z} = \\varphi F_{\\bar z} + \\psi G_{\\bar z}$ for a pseudoanalytic $w$.\nSince $\\varphi$ and $\\psi$ are real-linear in $w$, this is an equation $w_{\\bar z} = a w + b \\bar w$ with coefficients $a, b$ determined by $F$ and $G$.\nSome authors ask for less regularity of $\\varphi$ and $\\psi$ and use a weak form of the condition.\n",
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