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  "body": "\n## Statement\n\nLet $a \u003c 0 \u003c b$ and let $f \\colon [a, b] \\to \\C$ be [[Hölder continuous|0WZTM0EFR6]].\nThen the two limits below exist, and\n$$\\lim_{\\varepsilon \\to 0^+} \\int_a^b \\frac{f(x)}{x + i\\varepsilon} \\, dx = -i\\pi f(0) + \\mathrm{p.v.} \\int_a^b \\frac{f(x)}{x} \\, dx,$$\n$$\\lim_{\\varepsilon \\to 0^+} \\int_a^b \\frac{f(x)}{x - i\\varepsilon} \\, dx = i\\pi f(0) + \\mathrm{p.v.} \\int_a^b \\frac{f(x)}{x} \\, dx,$$\nwhere $\\mathrm{p.v.}$ denotes the [[Cauchy principal value|0V3RDQV3ZX]].\n\n## Remarks\n\nProof idea: write $\\frac{1}{x \\pm i\\varepsilon} = \\frac{x}{x^2 + \\varepsilon^2} \\mp i \\frac{\\varepsilon}{x^2 + \\varepsilon^2}$.\nThe second kernel, divided by $\\pi$, is an approximate identity, so its integral against $f$ tends to $\\pi f(0)$.\nThe first kernel, integrated against $f$, tends to the principal value; the Hölder condition makes $(f(x) - f(-x))/x$ integrable near $0$.\nIn the language of distributions, $\\frac{1}{x \\pm i0} = \\mathrm{p.v.} \\frac{1}{x} \\mp i\\pi \\delta$, where $\\delta$ is the [[Dirac delta|0KPHENZ5S7]].\nExample: take $f = 1$ on $[-1, 1]$ and let $\\operatorname{Log}$ be the principal logarithm.\nThe first integral is $\\operatorname{Log}(1 + i\\varepsilon) - \\operatorname{Log}(-1 + i\\varepsilon)$, which tends to $0 - i\\pi$, and the principal value is $0$.\nThere is a version for Cauchy-type integrals over a curve, the Sokhotski–Plemelj formulae, which give the jump of $\\frac{1}{2\\pi i} \\int_C \\frac{\\varphi(\\zeta)}{\\zeta - z} \\, d\\zeta$ across $C$.\n\n## Explanations\n\n### L1 human\n\nThe function $1/x$ cannot be integrated across $0$ in the usual sense.\nMoving the pole slightly off the real line, to $\\mp i\\varepsilon$, makes the integral finite.\nAs the pole moves back, the integral splits into a principal value plus a point contribution $\\mp i\\pi f(0)$.\nThe sign of that contribution records from which side the pole approached.\n\n### L2 human\n\nThe theorem gives the boundary values of Cauchy integrals, which is the starting point of the theory of singular integral equations and of the Riemann–Hilbert problem.\nIn physics it appears whenever one adds $i\\varepsilon$ to a denominator, as in Green's functions of quantum mechanics and in the [[Kramers–Kronig relations|GFS95PZ5YK]].\nIt is also the computation that relates the [[Hilbert transform|NSCEBXM7ZM]] to boundary values of holomorphic functions on a half-plane.\n",
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