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  "body": "\n## Definition\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]], let $M = \\max \\abs{K}$, and let $K_n$ be its [[iterated kernels|9QNBFERMHD]].\nFor $\\lambda \\in \\C$ with $\\abs{\\lambda} M (b - a) \u003c 1$, the *resolvent kernel* of $K$ is\n$$R(x, s; \\lambda) = \\sum_{n=1}^{\\infty} \\lambda^{n-1} K_n(x, s), \\quad x, s \\in [a, b].$$\nThe series converges absolutely and [[uniformly|08VZ57PT26]] on $[a, b] \\times [a, b]$, because $\\abs{\\lambda^{n-1} K_n} \\le M (\\abs{\\lambda} M (b - a))^{n-1}$.\n\n## Remarks\n\nExample: $K(x, s) = x s$ on $[0, 1]$, so $M = 1$ and $K_n = x s / 3^{n-1}$.\nFor $\\abs{\\lambda} \u003c 1$ the series is $x s \\sum_{n \\ge 1} (\\lambda/3)^{n-1} = 3 x s / (3 - \\lambda)$.\n\nNon-example: for $K = 1$ on $[0, 1]$ and $\\lambda = 1$, all $K_n = 1$ and the series $\\sum 1$ diverges.\nThis $\\lambda$ violates the condition $\\abs{\\lambda} M (b - a) \u003c 1$, and indeed the homogeneous equation $\\varphi = \\int_0^1 \\varphi$ has the nonzero solution $\\varphi = 1$.\n\nThe solution of the [[integral equation of the second kind|8KJCN49V64]] $\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds$ is then $\\varphi(x) = f(x) + \\lambda \\int_a^b R(x, s; \\lambda) f(s) \\, ds$.\nIn operator terms, $R = K (I - \\lambda K)^{-1}$.\nBy Fredholm theory, $R$ extends to a meromorphic function of $\\lambda$ on $\\C$, the quotient of two entire functions.\nThe resolvent kernel is then defined at every $\\lambda$ that is not a pole.\nThe example shows that the extension can go beyond the disc of the definition: $3 x s / (3 - \\lambda)$ has its only pole at $\\lambda = 3$.\n",
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