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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.\nLet $f \\colon [a, b] \\to \\C$ be a given function, and let $\\lambda \\in \\C$ be a parameter.\nLet $\\varphi$ denote an unknown function on $[a, b]$.\nThe *Fredholm equation of the second kind* with kernel $K$, parameter $\\lambda$ and right-hand side $f$ is\n$$\\varphi(x) - \\lambda \\int_a^b K(x, y) \\, \\varphi(y) \\, dy = f(x), \\qquad x \\in [a, b].$$\nIt is *homogeneous* when $f = 0$.\n\n## Remarks\n\nExample: on $[0, 1]$, take $K(x, y) = xy$, $\\lambda = 1$ and $f(x) = x$.\nThen $\\varphi(x) = 3x/2$ is a solution, since $\\int_0^1 x y \\cdot \\tfrac{3y}{2} \\, dy = \\tfrac{3x}{2} \\cdot \\tfrac{1}{3} = \\tfrac{x}{2}$ and $\\tfrac{3x}{2} - \\tfrac{x}{2} = x$.\n\nNon-example: $\\int_a^b K(x, y) \\varphi(y) \\, dy = f(x)$ is not of the second kind, because $\\varphi$ does not appear outside the integral; it is a [[Fredholm equation of the first kind|2RMWAVR1HV]].\n\nSome books write the equation as $\\varphi - \\lambda T_K \\varphi = f$, where $(T_K \\varphi)(x) = \\int_a^b K(x, y) \\varphi(y) \\, dy$.\nOthers put $\\lambda$ on the other side, as $\\lambda \\varphi - T_K \\varphi = f$; for $\\lambda \\ne 0$ the two forms are equivalent after dividing by $\\lambda$.\n",
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