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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]].\nFor points $z_1, \\dots, z_m$ and $w_1, \\dots, w_m$ of $[a, b]$, write\n$$K\\begin{pmatrix} z_1, \\dots, z_m \\\\ w_1, \\dots, w_m \\end{pmatrix} = \\det[K(z_i, w_j)]_{i,j=1}^{m},$$\nthe [[determinant|00KDZRW4C0]] of the $m \\times m$ matrix with entries $K(z_i, w_j)$.\nLet $n \\ge 1$, and let $x_1, \\dots, x_n, y_1, \\dots, y_n \\in [a, b]$.\nThe *Fredholm minor of order $n$* of $K$ is the function\n$$D_n\\begin{pmatrix} x_1, \\dots, x_n \\\\ y_1, \\dots, y_n \\end{pmatrix}(\\lambda) = \\sum_{k=0}^{\\infty} \\frac{(-\\lambda)^k}{k!} \\int_a^b \\cdots \\int_a^b K\\begin{pmatrix} x_1, \\dots, x_n, t_1, \\dots, t_k \\\\ y_1, \\dots, y_n, t_1, \\dots, t_k \\end{pmatrix} dt_1 \\cdots dt_k$$\nof $\\lambda \\in \\C$, where the term with $k = 0$ is $K\\begin{pmatrix} x_1, \\dots, x_n \\\\ y_1, \\dots, y_n \\end{pmatrix}$.\nThe minor of order $1$ is the *first minor*, written $D(x, y; \\lambda)$.\n\n## Remarks\n\nThe series converges for every $\\lambda$, uniformly in the points, by Hadamard's bound on determinants.\nSo each minor is an entire function of $\\lambda$ and a continuous function of the points.\nWith $n = 0$ the same formula gives the [[Fredholm determinant|PPX0AK81QV]] $D(\\lambda)$.\n\nExample: for $K(x, y) = xy$ on $[0, 1]$, every matrix $[z_i w_j]$ of size at least $2$ has rank $1$, so its determinant is $0$.\nHence $D(x, y; \\lambda) = xy$, and all minors of order $n \\ge 2$ vanish identically.\n\nNon-example: the series $\\sum_k \\frac{(-\\lambda)^k}{k!} \\int K\\begin{pmatrix} t_1, \\dots, t_k \\\\ t_1, \\dots, t_k \\end{pmatrix} dt$, with no fixed points, is not a minor of order $n \\ge 1$; it is the determinant $D(\\lambda)$ itself.\n\nSome authors put an extra factor $\\lambda^n$ in front of the minor of order $n$; the formulas that involve the minors change accordingly.\n",
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