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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{C}$ be an open set, and identify $\\mathbb{C}$ with $\\mathbb{R}^2$ by $x + iy \\leftrightarrow (x, y)$.\nLet $u : U \\to \\mathbb{R}$ be a harmonic function: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$.\nA harmonic conjugate of $u$ is a function $v : U \\to \\mathbb{R}$ such that the function $f = u + iv$ is [[holomorphic|0DG0Z72CQD]] on $U$.\n\n## Remarks\n\nFor example, $v = 2xy$ is a harmonic conjugate of $u = x^2 - y^2$ on $\\mathbb{C}$, because $u + iv = z^2$.\nIf $U$ is a [[domain|097KNJFJ7V]], two harmonic conjugates of $u$ differ by a real constant.\nIf $v$ is a harmonic conjugate of $u$, then $-u$ is a harmonic conjugate of $v$, because $v - iu = -i(u + iv)$ is holomorphic.\nOn a disc, each harmonic function has a harmonic conjugate. On $\\mathbb{C} \\setminus \\{0\\}$, the harmonic function $\\log |z|$ has none.\n\n## Explanations\n\n### L1 human\n\nThe [[real part of a holomorphic function is harmonic|031BG625Y3]]. A harmonic conjugate goes the other way: it starts with a harmonic function $u$ and finds the missing imaginary part $v$.\nFor $u = x$, the conjugate is $v = y$, because $x + iy = z$ is holomorphic.\n",
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