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  "body": "\n## Definition\n\nLet $P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\partial^\\alpha$ be a [[linear differential operator|6DZETZ5T9G]] on $\\R^n$ with smooth coefficients.\nA distribution is a continuous linear functional on the space of [[test functions|HS65DW0HXH]] on $\\R^n$.\nLet $\\delta$ be the [[Dirac delta|0KPHENZ5S7]] at the origin, the distribution $\\delta(\\varphi) = \\varphi(0)$.\nA *fundamental solution* of $P$ is a distribution $E$ on $\\R^n$ with $P E = \\delta$ in the sense of distributions.\nExplicitly, this means $E\\left(P^{t} \\varphi\\right) = \\varphi(0)$ for every test function $\\varphi$, where $P^{t} \\varphi = \\sum_{\\abs{\\alpha} \\le m} (-1)^{\\abs{\\alpha}} \\partial^\\alpha (a_\\alpha \\varphi)$ is the formal transpose of $P$.\n\n## Remarks\n\nExample: on $\\R$, the Heaviside function $H$, with $H(x) = 1$ for $x \u003e 0$ and $H(x) = 0$ for $x \u003c 0$, is a fundamental solution of $P = d/dx$.\nIndeed, for a test function $\\varphi$ one has $H(-\\varphi') = -\\int_0^\\infty \\varphi'(x) \\, dx = \\varphi(0)$.\nExample: on $\\R^3$, $E(x) = -1/(4 \\pi \\abs{x})$ is a fundamental solution of the [[Laplacian|0E0B6RA8K0]], and on $\\R^2$, $E(x) = \\frac{1}{2\\pi} \\log \\abs{x}$ is one.\nNon-example: on $\\R$, the constant function $1$ is not a fundamental solution of $d/dx$, since its derivative is $0$ and not $\\delta$.\nA fundamental solution is not unique: adding any distribution $F$ with $P F = 0$ gives another one.\nIf $P$ has constant coefficients and $f$ is a test function, then $u = E * f$ is smooth and solves $P u = f$.\nEvery nonzero operator with constant coefficients has a fundamental solution, by the [[Malgrange–Ehrenpreis theorem|3173Y813V7]].\n\n## Explanations\n\n### L1 human\n\nA fundamental solution is a solution of the equation with a unit point source at the origin as right-hand side.\n",
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