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  "body": "\n## Statement\n\nLet $H = \\set{z \\in \\C : \\operatorname{Im} z \u003e 0}$ be the upper half-plane.\nLet $\\chi$ be continuous on the closure $\\bar H$ and [[holomorphic|0DG0Z72CQD]] on $H$, with $\\chi(z) \\to 0$ as $\\abs{z} \\to \\infty$ in $\\bar H$, uniformly in the direction.\nWrite $\\chi(\\omega) = \\chi_1(\\omega) + i \\chi_2(\\omega)$ for real $\\omega$, with $\\chi_1$ and $\\chi_2$ real.\nThen for each real $\\omega$,\n$$\\chi_1(\\omega) = \\frac{1}{\\pi} \\, \\mathrm{p.v.} \\int_{-\\infty}^{\\infty} \\frac{\\chi_2(\\omega')}{\\omega' - \\omega} \\, d\\omega', \\qquad \\chi_2(\\omega) = -\\frac{1}{\\pi} \\, \\mathrm{p.v.} \\int_{-\\infty}^{\\infty} \\frac{\\chi_1(\\omega')}{\\omega' - \\omega} \\, d\\omega'.$$\nHere $\\mathrm{p.v.}$ is the [[principal value|0V3RDQV3ZX]] at $\\omega$ and at infinity: the limit of the integral over $\\varepsilon \u003c \\abs{\\omega' - \\omega} \u003c R$ as $\\varepsilon \\to 0$ and $R \\to \\infty$.\n\n## Remarks\n\nProof idea: integrate $\\chi(z)/(z - \\omega)$ over the boundary of the half-disc of radius $R$ in $H$, with a small half-circle of radius $\\varepsilon$ around $\\omega$ removed.\nBy [[Cauchy's theorem|01MPZXYVYF]] the integral is $0$.\nThe big half-circle contributes at most $\\pi \\max \\abs{\\chi} \\cdot R/(R - \\abs{\\omega}) \\to 0$, and the small one tends to $-i\\pi \\chi(\\omega)$.\nSo $\\mathrm{p.v.} \\int \\frac{\\chi(\\omega')}{\\omega' - \\omega} \\, d\\omega' = i\\pi \\chi(\\omega)$, and the real and imaginary parts give the two relations.\nExample: $\\chi(z) = \\frac{1}{z + i}$ gives $\\chi_1(\\omega) = \\frac{\\omega}{\\omega^2 + 1}$ and $\\chi_2(\\omega) = -\\frac{1}{\\omega^2 + 1}$, and these satisfy both relations.\nIn terms of the [[Hilbert transform|NSCEBXM7ZM]], each of $\\chi_1$ and $\\chi_2$ determines the other.\n\n## Explanations\n\n### L1 human\n\nIn physics, a response function such as the refractive index of a material has a real part and an imaginary part.\nCausality, the fact that an effect cannot come before its cause, makes the response holomorphic in the upper half-plane of frequencies.\nThen the real part and the imaginary part are not independent: each one can be computed from the other by an integral.\nFor example, the absorption of light at all frequencies determines the refractive index.\n\n### L2 human\n\nThe relations are a direct consequence of Cauchy's integral formula on a half-plane.\nThey say that the real and imaginary parts of the boundary values of a holomorphic function on $H$ that decays at infinity are Hilbert transforms of each other.\nFor square-integrable boundary values, Titchmarsh's theorem makes this an equivalence with the Hardy space $H^2$ of the upper half-plane.\nThey are used to check measured spectra, to recover phases from amplitudes, and as dispersion relations in particle physics.\n",
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