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  "body": "\n## Summary\n\nLet $f$ be an integrable function on the circle, and extend it to a harmonic function in the unit disc.\nThe conjugate function $\\tilde f$ is the boundary value of the [[harmonic conjugate|050F2FMTKF]] of this extension.\nOn Fourier series it acts by $e^{ik\\theta} \\mapsto -i \\sgn(k) e^{ik\\theta}$, so it turns $\\cos k\\theta$ into $\\sin k\\theta$.\nIt is the circle version of the [[Hilbert transform|NSCEBXM7ZM]].\nIts $L^p$ bounds control the convergence of [[Fourier series|G12KXDHPPS]] in $L^p$.\n\n## Explanations\n\n### L1 human\n\nEvery real function $f$ on the circle is the real part of the boundary values of some holomorphic function in the disc.\nThe imaginary part of these boundary values is the conjugate function $\\tilde f$.\nIn terms of waves, it shifts every cosine wave into the matching sine wave, and every sine wave into minus the cosine wave.\nFor example, the holomorphic function $z = re^{i\\theta}$ equals $r\\cos\\theta + i r\\sin\\theta$.\nOn the circle $r = 1$ its real part is $\\cos\\theta$ and its imaginary part is $\\sin\\theta$, so the conjugate of $\\cos\\theta$ is $\\sin\\theta$.\n",
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  "name": "Conjugate function",
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