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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\Omega \\subseteq \\R^n$ be an open set, and let $k \u003e 0$ be a real constant.\nLet $u \\colon \\Omega \\to \\C$ be twice differentiable.\nThe *Helmholtz equation* on $\\Omega$ with wave number $k$ is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\Delta u + k^2 u = 0,$$\nwhere $\\Delta = \\partial_1^2 + \\cdots + \\partial_n^2$ is the [[Laplacian|0E0B6RA8K0]].\nThe *inhomogeneous Helmholtz equation* is $\\Delta u + k^2 u = -f$ for a given function $f$ on $\\Omega$.\n\n## Remarks\n\nExample: in $\\R^n$, for a unit vector $\\omega$, the plane wave $u(x) = e^{i k \\, \\omega \\cdot x}$ is a solution, since $\\Delta u = (ik)^2 \\abs{\\omega}^2 u = -k^2 u$.\nNon-example: $u(x) = e^{k x_1}$ is not a solution, since $\\Delta u + k^2 u = 2k^2 u \\ne 0$; it solves instead $\\Delta u - k^2 u = 0$, the modified Helmholtz equation.\nIf $v(x, t) = u(x) e^{-i \\omega t}$ solves the wave equation $\\partial_t^2 v = c^2 \\Delta v$, then $u$ solves the Helmholtz equation with $k = \\omega / c$.\nOn a bounded domain with zero boundary values, the equation has a nonzero solution exactly when $k^2$ is an eigenvalue of $-\\Delta$ with that boundary condition.\nSome authors allow complex $k$.\n",
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