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  "body": "\n## Definition\n\nLet $(x_n)_{n \\in \\Z}$ or $(x_n)_{n \\ge 0}$ be a [[sequence|0G9EKM0E15]] of complex numbers.\nThe *unilateral Z-transform* of $x$ is\n$$X(z) = \\sum_{n = 0}^{\\infty} x_n z^{-n},$$\nas a function of the nonzero complex numbers $z$ for which the series converges absolutely.\n\n## Remarks\n\nLet $r = \\limsup_{n \\to \\infty} \\abs{x_n}^{1/n}$.\nThe series converges absolutely for $\\abs{z} \u003e r$ and diverges for $0 \u003c \\abs{z} \u003c r$, by the root test.\nExample: $x_n = n$ gives $X(z) = \\sum_{n \\ge 0} n z^{-n} = z/(z - 1)^2$ for $\\abs{z} \u003e 1$, by differentiating the geometric series.\nNon-example: $x_n = n!$ gives $r = \\infty$, so the series converges for no $z$, and $x$ has no unilateral Z-transform as a function.\nThe unilateral transform of $x$ equals the [[Z-transform|5FY0JQGHYZ]] of the sequence that agrees with $x$ for $n \\ge 0$ and is $0$ for $n \u003c 0$.\nWriting $w = z^{-1}$ turns it into the ordinary generating function $\\sum_{n \\ge 0} x_n w^n$.\n",
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    "name": "Unilateral Z-transform",
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