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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers.\nLet $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[kernel|GSMBXT7FPK]], for example continuous or square-integrable, and let $f \\colon [a, b] \\to \\C$ be a given function.\nLet $\\varphi$ denote an unknown function on $[a, b]$.\nThe *Fredholm equation of the first kind* with kernel $K$ and right-hand side $f$ is\n$$\\int_a^b K(x, y) \\, \\varphi(y) \\, dy = f(x), \\qquad x \\in [a, b].$$\nA solution is a function $\\varphi$ in the chosen function space for which the integral exists and the equality holds.\nIn $L^2$ the equality is required for almost every $x$, and otherwise for every $x$.\n\n## Remarks\n\nExample: on $[0, 1]$, take $K(x, y) = xy$ and $f(x) = x / 2$.\nThe constant function $\\varphi = 1$ is a solution, because $\\int_0^1 x y \\, dy = x/2$ for every $x$.\nIt is not the only one: $\\varphi(y) = 1 + (2 - 3y)$ also works, since $\\int_0^1 y (2 - 3y) \\, dy = 1 - 1 = 0$.\nWith the same kernel and $f(x) = 1$, there is no solution, since the left side is always a multiple of $x$ and vanishes at $x = 0$.\n\nNon-example: the equation $\\varphi(x) - \\int_a^b K(x, y) \\varphi(y) \\, dy = f(x)$ is not of the first kind, because $\\varphi$ also appears outside the integral; it is a Fredholm equation of the second kind.\n\nThe equation is a special [[Fredholm integral equation|526TCVMSHP]]: the limits of integration are the fixed numbers $a$ and $b$.\nIn operator notation it reads $T_K \\varphi = f$, where $(T_K \\varphi)(x) = \\int_a^b K(x, y) \\varphi(y) \\, dy$.\n",
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