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  "body": "\n## Summary\n\nThe resolvent kernel $R(x, s; \\lambda)$ of a [[kernel|GSMBXT7FPK]] $K$ gives the solution of the [[integral equation of the second kind|8KJCN49V64]] $\\varphi = f + \\lambda K \\varphi$.\nThe solution is $\\varphi(x) = f(x) + \\lambda \\int_a^b R(x, s; \\lambda) f(s) \\, ds$.\nFor small $\\abs{\\lambda}$ it is the sum $\\sum_{n \\ge 1} \\lambda^{n-1} K_n$ of the [[iterated kernels|9QNBFERMHD]].\nIt thus inverts the operator $I - \\lambda K$ once and for all, for every right side $f$.\nFor a continuous kernel it extends to a meromorphic function of $\\lambda$, whose poles are characteristic values of $K$.\n\n## Explanations\n\n### L1 human\n\nSolving the equation for each new right side separately is wasteful.\nThe resolvent kernel is a single formula that turns any right side directly into the solution.\nIt plays the role of the inverse matrix in linear algebra.\nFor example, for $K(x, s) = x s$ on $[0, 1]$ the resolvent kernel is $R(x, s; \\lambda) = 3 x s / (3 - \\lambda)$.\n",
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