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  "body": "\n## Definition\n\nLet $a \u003c b$, let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[kernel|GSMBXT7FPK]], let $f \\colon [a, b] \\to \\C$ be given, and let $\\lambda \\in \\C$.\nA *Fredholm integral equation of the first kind* is the equation\n$$\\int_a^b K(x, s) \\varphi(s) \\, ds = f(x) \\quad \\text{for all } x \\in [a, b].$$\nA *Fredholm integral equation of the second kind* is the equation\n$$\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds \\quad \\text{for all } x \\in [a, b].$$\nIn both, $\\varphi \\colon [a, b] \\to \\C$ is the unknown function, and both limits of integration are the fixed numbers $a$ and $b$.\nA *Fredholm integral equation* is a [[linear integral equation|AYZCKG34QK]] of one of these two forms.\n\n## Remarks\n\nExample: $\\varphi(x) = x + \\int_0^1 x s \\, \\varphi(s) \\, ds$ on $[0, 1]$ is of the second kind.\nA solution has the form $\\varphi(x) = c x$ with $c = 1 + c/3$, so $\\varphi(x) = 3x/2$.\n\nNon-example: $\\varphi(x) = x + \\int_0^x x s \\, \\varphi(s) \\, ds$ is not a Fredholm equation, since the upper limit is the variable $x$; it is a Volterra equation.\n\nA Volterra equation can be written as a Fredholm equation with the kernel set to $0$ for $s \u003e x$, but this kernel is in general discontinuous on the diagonal.\nThe same definitions are used with $[a, b]$ replaced by a region of $\\R^m$ or a measure space.\n",
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