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  "body": "\n## Definition\n\nLet $x = (x_n)_{n \\in \\Z}$ be a two-sided [[sequence|0G9EKM0E15]] of complex numbers.\nThe *bilateral Z-transform* of $x$ is the [[Laurent series|0GXMPDNYX3]]\n$$X(z) = \\sum_{n = -\\infty}^{\\infty} x_n z^{-n}.$$\nIt is a function on its [[region of convergence|1P8WX3EGHF]], the set of nonzero complex $z$ where the series converges absolutely.\n\n## Remarks\n\nExample: let $x_n = 2^{-\\abs{n}}$.\nThe part $\\sum_{n \\ge 0} (2z)^{-n}$ converges for $\\abs{z} \u003e 1/2$ and the part $\\sum_{n \\ge 1} (z/2)^n$ converges for $\\abs{z} \u003c 2$.\nSo $X(z) = \\dfrac{2z}{2z - 1} + \\dfrac{z}{2 - z}$ on the annulus $1/2 \u003c \\abs{z} \u003c 2$.\nExample with the same formula: for $a \\ne 0$, $x_n = a^n$ for $n \\ge 0$ (and $0$ otherwise) gives $z/(z - a)$ on $\\abs{z} \u003e \\abs{a}$, while $x_n = -a^n$ for $n \\le -1$ (and $0$ otherwise) gives $z/(z - a)$ on $\\abs{z} \u003c \\abs{a}$.\nNon-example: $x_n = 1$ for all $n$ has an empty region of convergence, so the transform is not a function.\nThe notation $X(z) = \\mathcal{Z}\\{x\\}(z)$ is standard; signal processing texts write $x[n]$ for $x_n$.\n",
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