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  "body": "\n## Example\n\nLet $\\Gamma \\subset \\R^2$ be a closed $C^2$ curve, and let $f$ be a continuous function on $\\Gamma$.\nSymm's integral equation asks for a density $\\psi$ on $\\Gamma$ with\n$$-\\frac{1}{2\\pi} \\int_\\Gamma \\ln \\abs{x - y} \\, \\psi(y) \\, ds(y) = f(x), \\quad x \\in \\Gamma.$$\nThe left side is the [[single layer potential|9T6ZEV4125]] of $\\psi$ in the plane, a [[logarithmic potential|46NHWJHW44]], restricted to $\\Gamma$.\nThe unknown appears only under the integral, so it is an [[integral equation of the first kind|ZDS77Q0F06]], with the [[weakly singular kernel|94TNBAAJYX]] $-\\frac{1}{2\\pi}\\ln\\abs{x - y}$.\nSolving it gives a single layer potential that solves the interior Dirichlet problem for the Laplace equation with boundary values $f$.\n\nCheck on the circle $\\Gamma = \\set{x : \\abs{x} = a}$, with $y = a e^{i\\theta}$ and $x = a e^{i\\phi}$:\nby the mean value property, $\\frac{1}{2\\pi} \\int_0^{2\\pi} \\ln \\abs{x - a e^{i\\theta}} \\, d\\theta = \\ln a$ for $\\abs{x} = a$, and $\\ln \\abs{e^{i\\phi} - e^{i\\theta}} = -\\sum_{n \\ge 1} \\frac{\\cos n(\\phi - \\theta)}{n}$.\nHence the operator maps $\\psi = 1$ to the constant $-a \\ln a$, and $\\psi(\\theta) = e^{in\\theta}$ with $n \\ne 0$ to $\\frac{a}{2\\abs{n}} e^{in\\phi}$.\nSo on a circle of radius $a \\ne 1$ the equation is uniquely solvable for every trigonometric polynomial $f$.\nFor $a = 1$ the constant density $\\psi = 1$ solves the homogeneous equation.\n\n## Remarks\n\nThe circle example shows the general picture: the homogeneous equation has only the zero solution unless $\\Gamma$ has logarithmic capacity $1$, and this exceptional case can be avoided by rescaling.\nThe eigenvalues $\\frac{a}{2\\abs{n}}$ tend to $0$, so the solution depends unstably on $f$ in the supremum norm, as for every first-kind equation with a compact operator.\nBecause of its simple structure, the equation is a standard test case for boundary element and collocation methods.\n\n## Explanations\n\n### L1 human\n\nPut an unknown charge density on a closed curve in the plane, and ask that its potential takes prescribed values on the curve.\nThat is Symm's equation.\nOn a circle everything can be computed with Fourier series: each wave pattern of the charge is just multiplied by a number.\n",
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