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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]], and let $\\lambda \\in \\C$.\nConsider the [[integral equation of the second kind|8KJCN49V64]]\n$$\\varphi(x) - \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds = f(x), \\quad x \\in [a, b].$$\nIts *adjoint kernel* is $K^*(x, s) = \\overline{K(s, x)}$, where the bar is the [[complex conjugate|0FTH2E9TAS]].\nFor a given function $g \\colon [a, b] \\to \\C$, the *adjoint integral equation* is\n$$\\psi(x) - \\bar\\lambda \\int_a^b \\overline{K(s, x)} \\, \\psi(s) \\, ds = g(x), \\quad x \\in [a, b],$$\nfor the unknown function $\\psi \\colon [a, b] \\to \\C$.\n\n## Remarks\n\nExample: for $K(x, s) = i x$ on $[0, 1]$, the adjoint kernel is $K^*(x, s) = \\overline{i s} = -i s$.\nThe check: put $x$ and $s$ in reverse order and conjugate.\n\nNon-example: the equation with kernel $K(s, x)$ and parameter $\\lambda$, without conjugation, is the adjoint only when $K$ is real and $\\lambda$ is real.\nFor $K(x, s) = i x$ it gives the kernel $i s$, not $-i s$.\n\nWith the inner product $\\ip{u}{v} = \\int_a^b u(x) \\overline{v(x)} \\, dx$, the integral operators satisfy $\\ip{K u}{v} = \\ip{u}{K^* v}$ when $K$ is continuous or square-integrable.\nFor real kernels and real $\\lambda$, the adjoint equation is often called the *transposed* or *associated* equation.\nA kernel equal to its adjoint kernel is called Hermitian.\n",
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