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  "body": "\n## Definition\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$ be the circle, and let $f \\colon \\mathbb{T} \\to \\C$ be [[Lebesgue integrable|DWJF7ABPR1]].\nThe *conjugate function* of $f$ is the function $\\tilde f$ on $\\mathbb{T}$ given by the [[principal value|0SYR10HD4R]]\n$$\\tilde f(\\theta) = \\lim_{\\varepsilon \\to 0^+} \\frac{1}{2\\pi} \\int_{\\varepsilon \\le \\abs{t} \\le \\pi} f(\\theta - t) \\cot\\frac{t}{2} \\, dt.$$\nThe limit exists for almost every $\\theta \\in \\mathbb{T}$.\n\n## Remarks\n\nIf $f$ is a trigonometric polynomial with [[Fourier series|G12KXDHPPS]] $\\sum_k \\hat f(k) e^{ik\\theta}$, then\n$$\\tilde f(\\theta) = \\sum_k (-i \\sgn k) \\, \\hat f(k) e^{ik\\theta},$$\nwith $\\sgn 0 = 0$.\nFor real $f$, let $u$ be the Poisson integral of $f$ in the unit disc, and let $v$ be its [[harmonic conjugate|050F2FMTKF]] with $v(0) = 0$.\nThen $v(re^{i\\theta}) \\to \\tilde f(\\theta)$ as $r \\to 1^-$ for almost every $\\theta$.\n\nExample: $f(\\theta) = \\cos\\theta$ has $\\tilde f(\\theta) = \\sin\\theta$.\nCheck: $\\cos(\\theta - t)\\cot(t/2) = \\cos\\theta \\cos t \\cot(t/2) + \\sin\\theta \\sin t \\cot(t/2)$.\nThe first term is odd in $t$, so its principal value integral is $0$.\nThe second uses $\\sin t \\cot(t/2) = 2\\cos^2(t/2) = 1 + \\cos t$, whose integral over $[-\\pi, \\pi]$ is $2\\pi$.\n\nNon-example: the integral without the principal value need not exist.\nFor $f = 1$ near $\\theta$ the integrand behaves like $2/t$ near $t = 0$, which is not integrable, while the symmetric limit is $0$ since $\\cot(t/2)$ is odd.\n\nThe function $\\tilde f$ need not be integrable when $f$ is: the operator $f \\mapsto \\tilde f$ is bounded on $L^p(\\mathbb{T})$ only for $1 \u003c p \u003c \\infty$.\n",
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