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  "body": "\n## Statement\n\nLet $x = (x_n)_{n \\in \\Z}$ be a sequence of complex numbers whose [[bilateral Z-transform|3DBAMXA80B]] $X(z) = \\sum_{n \\in \\Z} x_n z^{-n}$ has a [[region of convergence|1P8WX3EGHF]] that contains a circle $\\abs{z} = \\rho$ with $\\rho \u003e 0$.\nThen for every $n \\in \\Z$,\n$$x_n = \\frac{1}{2\\pi i} \\oint_{\\abs{z} = \\rho} X(z) \\, z^{n-1} \\, dz,$$\nwhere the [[contour integral|06ZGPYTWK8]] runs once counterclockwise around the circle.\nIn particular, $x$ is determined by the function $X$ on any one such circle.\n\n## Remarks\n\nOn the circle the series $\\sum_k x_k z^{-k}$ converges absolutely and uniformly, so it can be integrated term by term after multiplication by $z^{n-1}$.\nThe integral of $z^{n-1-k}$ around the circle is $2\\pi i$ when $k = n$ and $0$ otherwise, which leaves $x_n$.\nThis is the coefficient formula for the [[Laurent series|0GXMPDNYX3]] of $X$ on its annulus.\nExample: $X(z) = z/(z - a)$ on $\\abs{z} \u003e \\abs{a}$; for $n \\ge 0$ the residue theorem gives $x_n = a^n$, and for $n \u003c 0$ the integrand $z^n/(z - a)$ has total residue $0$ inside a large circle, so $x_n = 0$.\nThe circle must lie in the region of convergence: the same rational function $z/(z - a)$ on $\\abs{z} \u003c \\abs{a}$ gives $x_n = -a^n$ for $n \\le -1$ and $x_n = 0$ for $n \\ge 0$.\n\n## Explanations\n\n### L1 human\n\nThe Z-transform packs a sequence into a function, and this formula unpacks it.\nEach term of the sequence is recovered by an average of the function around a circle.\nThe circle must be chosen inside the region where the transform converges, since that region tells which sequence one started from.\n\n### L2 human\n\nIn practice, $X$ is often a rational function, and one inverts it by partial fractions and a table of pairs such as $a^n \\leftrightarrow z/(z - a)$.\nThe contour formula justifies this method and shows that the result is unique once the region of convergence is fixed.\nWith $z = e^{i\\omega}$ on the unit circle, the formula becomes the inverse of the discrete-time Fourier transform.\nIt is the discrete analogue of the Bromwich inversion integral for the Laplace transform.\n",
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  "msc": [
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  "name": "Inverse Z-transform",
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