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  "body": "\n## Summary\n\nThe Fredholm determinant of a continuous kernel $K$ on $[a, b]$ is the entire function $D(\\lambda) = \\sum_{n \\ge 0} \\frac{(-\\lambda)^n}{n!} \\int \\cdots \\int \\det[K(x_i, x_j)] \\, dx_1 \\cdots dx_n$.\nIt plays for the integral operator $\\varphi \\mapsto \\int K(\\cdot, y) \\varphi(y) \\, dy$ the role that $\\det(I - \\lambda A)$ plays for a square matrix $A$.\nIts zeros are the [[characteristic values|DC7HMHVREP]] of the kernel.\nTogether with the [[Fredholm minors|HP5VJ9EBST]] it gives an explicit formula for the solution of the [[Fredholm equation of the second kind|DEYYNVCV9J]].\nWhen the integral operator $T$ of $K$ is a [[trace class operator|Z2CP530WDE]], $D(\\lambda)$ equals the operator determinant $\\det(I - \\lambda T)$.\n\n## Explanations\n\n### L1 human\n\nReplace the integral by a sum over $n$ equally spaced points, and the integral operator becomes an $n \\times n$ matrix.\nFor that matrix one can compute $\\det(I - \\lambda A)$ by expanding it into sums of principal minors.\nLetting the number of points grow turns these sums into multiple integrals, and the result is the Fredholm determinant.\nFor the kernel $K(x, y) = xy$ on $[0, 1]$ only the first two terms survive, and $D(\\lambda) = 1 - \\lambda/3$.\nIt vanishes at $\\lambda = 3$, which is exactly where the equation $\\varphi = \\lambda \\int_0^1 x y \\varphi(y) \\, dy$ has the nonzero solution $\\varphi(x) = x$.\n",
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