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  "body": "\n## Summary\n\nThe unilateral Z-transform of a sequence $(x_n)$ uses only its terms with $n \\ge 0$: $X(z) = \\sum_{n \\ge 0} x_n z^{-n}$.\nIt is the form of the [[Z-transform|5FY0JQGHYZ]] used for initial value problems.\nA forward shift of the sequence brings in its first terms, so initial values enter the transformed equation by themselves.\nIt plays for difference equations the part that the one-sided Laplace transform plays for differential equations.\n\n## Explanations\n\n### L1 human\n\nMany processes start at a time zero, and only the values from then on matter.\nThe unilateral Z-transform ignores everything before time zero.\nFor $x_n = 2^n$ it gives $\\sum_{n \\ge 0} 2^n z^{-n} = z/(z - 2)$, valid when $\\abs{z} \u003e 2$.\nIts set of convergence is always the outside of a circle centred at $0$, possibly together with some points of that circle.\nIt can also be empty or the whole plane without $0$.\n",
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