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  "body": "\n## Statement\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]] with $M = \\max \\abs{K}$, and let $\\rho = 1 / (M (b - a))$.\nFor $\\abs{\\lambda} \u003c \\rho$ let $R(x, s; \\lambda)$ be the [[resolvent kernel|ARTJV1XCRX]] of $K$.\nThen the following hold.\n\n1. For $\\abs{\\lambda} \u003c \\rho$ and all $x, s \\in [a, b]$,\n$$R(x, s; \\lambda) - K(x, s) = \\lambda \\int_a^b K(x, t) R(t, s; \\lambda) \\, dt = \\lambda \\int_a^b R(x, t; \\lambda) K(t, s) \\, dt.$$\n2. For $\\abs{\\lambda} \u003c \\rho$, $\\abs{\\mu} \u003c \\rho$ and all $x, s \\in [a, b]$,\n$$R(x, s; \\lambda) - R(x, s; \\mu) = (\\lambda - \\mu) \\int_a^b R(x, t; \\lambda) R(t, s; \\mu) \\, dt.$$\n\n## Remarks\n\nProof idea: write $R = \\sum_{n \\ge 1} \\lambda^{n-1} K_n$ with the [[iterated kernels|9QNBFERMHD]] and use $K_{n+1}(x, s) = \\int K(x, t) K_n(t, s) \\, dt = \\int K_n(x, t) K(t, s) \\, dt$; uniform convergence allows integration term by term.\nFor part 2, multiply out the two series and use $K_{n+m}(x, s) = \\int K_n(x, t) K_m(t, s) \\, dt$; the coefficient identity is $\\lambda^{k} - \\mu^{k} = (\\lambda - \\mu) \\sum_{i + j = k - 1} \\lambda^i \\mu^j$.\nIn operator form, with $R_\\lambda = K (I - \\lambda K)^{-1}$, the identities read $R_\\lambda - K = \\lambda K R_\\lambda = \\lambda R_\\lambda K$ and $R_\\lambda - R_\\mu = (\\lambda - \\mu) R_\\lambda R_\\mu$.\n\nExample: for $K(x, s) = x s$ on $[0, 1]$, $R(x, s; \\lambda) = 3 x s / (3 - \\lambda)$.\nThen $\\lambda \\int_0^1 x t \\cdot 3 t s / (3 - \\lambda) \\, dt = \\lambda x s / (3 - \\lambda)$, and $3 x s / (3 - \\lambda) - x s = \\lambda x s / (3 - \\lambda)$, as part 1 says.\n\nBoth identities stay true, by analytic continuation, for all $\\lambda$ and $\\mu$ that are not poles of the meromorphic extension of $R$.\n\n## Explanations\n\n### L1 human\n\nThe resolvent kernel solves an integral equation of its own, with the kernel $K$ itself as the free term.\nA second identity compares the resolvent kernels at two values of the parameter.\nThese rules let one compute with resolvent kernels as one computes with inverse matrices.\n\n### L2 human\n\nPart 1 is the kernel form of $(I + \\lambda R_\\lambda)(I - \\lambda K) = (I - \\lambda K)(I + \\lambda R_\\lambda) = I$, which says that $I + \\lambda R_\\lambda$ inverts $I - \\lambda K$.\nIt is often taken as the definition of the resolvent kernel beyond the disc of convergence of the series.\nPart 2 is the analogue of the resolvent identity of operator theory, and it shows that $R$ is analytic in $\\lambda$ with $\\partial_\\lambda R(x, s; \\lambda) = \\int_a^b R(x, t; \\lambda) R(t, s; \\lambda) \\, dt$.\n",
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