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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]].\nLet $D(\\lambda)$ be its [[Fredholm determinant|PPX0AK81QV]] and $D(x, y; \\lambda)$ its first [[Fredholm minor|HP5VJ9EBST]].\nLet $\\lambda \\in \\C$ with $D(\\lambda) \\ne 0$, and put\n$$R(x, y; \\lambda) = \\frac{D(x, y; \\lambda)}{D(\\lambda)}, \\qquad x, y \\in [a, b].$$\nThen:\n\n1. $R(\\cdot, \\cdot; \\lambda)$ is continuous on $[a, b]^2$ and satisfies\n$$R(x, y; \\lambda) - K(x, y) = \\lambda \\int_a^b K(x, t) R(t, y; \\lambda) \\, dt = \\lambda \\int_a^b R(x, t; \\lambda) K(t, y) \\, dt;$$\n2. for every continuous $f \\colon [a, b] \\to \\C$, the [[Fredholm equation of the second kind|DEYYNVCV9J]] $\\varphi(x) - \\lambda \\int_a^b K(x, y) \\varphi(y) \\, dy = f(x)$ has exactly one continuous solution, namely\n$$\\varphi(x) = f(x) + \\lambda \\int_a^b R(x, y; \\lambda) f(y) \\, dy;$$\n3. $\\lambda$ is not a [[characteristic value|DC7HMHVREP]] of $K$.\n\nSo $R$ is the [[resolvent kernel|ARTJV1XCRX]] of $K$, and it is a meromorphic function of $\\lambda$ whose poles lie among the zeros of $D$.\n\n## Remarks\n\nThe proof rests on two identities between entire functions of $\\lambda$.\nThey come from expanding the determinants in the series along the first row or the first column:\n$$D(x, y; \\lambda) = K(x, y) D(\\lambda) + \\lambda \\int_a^b K(x, t) D(t, y; \\lambda) \\, dt,$$\nand the same with $K$ and $D$ exchanged in the integral.\nDividing by $D(\\lambda)$ gives part 1, and part 1 shows by direct substitution that the formula in part 2 solves the equation.\nUniqueness comes from the same identities applied to a solution of the homogeneous equation, which must then be $0$.\n\nExample: for $K(x, y) = xy$ on $[0, 1]$, $D(\\lambda) = 1 - \\lambda/3$ and $D(x, y; \\lambda) = xy$.\nSo $R(x, y; \\lambda) = \\frac{3xy}{3 - \\lambda}$ for $\\lambda \\ne 3$.\nFor $|\\lambda| \u003c 3$ this agrees with the sum $\\sum_{n \\ge 0} \\lambda^n xy / 3^n$ of the [[Liouville–Neumann series|H1HWSWME6B]].\n\nThe Neumann series converges only for small $\\abs{\\lambda}$, while Fredholm's formula is valid for every $\\lambda$ with $D(\\lambda) \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nThis is Cramer's rule for integral equations.\nThe solution of a linear system is a ratio of determinants, and here the solution kernel is a ratio of two everywhere convergent power series in $\\lambda$.\nThe formula works as long as the denominator does not vanish.\n\n### L2 human\n\nThe result turns the [[Fredholm equation of the second kind|DEYYNVCV9J]] into an explicit computation whenever $D(\\lambda) \\ne 0$.\nIt extends the [[resolvent kernel|ARTJV1XCRX]] beyond the disc where the Neumann series converges, as a meromorphic function of $\\lambda$.\nAt a zero $\\lambda_0$ of $D$, the homogeneous equation has nonzero solutions, and these can be read off from the first Fredholm minor that does not vanish identically at $\\lambda_0$.\nIn operator language, $R$ is the kernel of $\\lambda^{-1}((I - \\lambda T_K)^{-1} - I)$, where $T_K$ is the integral operator with kernel $K$.\n",
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