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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $x \\colon \\Z^n \\to \\C$ be a sequence, written $x[k]$ for $k = (k_1, \\dots, k_n)$.\nThe *$n$-dimensional Z-transform* of $x$ is the function\n$$X(z_1, \\dots, z_n) = \\sum_{k \\in \\Z^n} x[k]\\, z_1^{-k_1} \\cdots z_n^{-k_n},$$\ndefined at those $(z_1, \\dots, z_n) \\in (\\C \\setminus \\set{0})^n$ where the series converges absolutely.\nThis set is its [[region of convergence|1P8WX3EGHF]].\n\n## Remarks\n\nFor $n = 1$ this is the bilateral [[Z-transform|5FY0JQGHYZ]].\nThe series is a [[Laurent series|0GXMPDNYX3]] in several variables.\nAbsolute convergence at $(z_1, \\dots, z_n)$ depends only on $(\\abs{z_1}, \\dots, \\abs{z_n})$.\n\nExample: $x[k_1, k_2] = a^{k_1} b^{k_2}$ for $k_1, k_2 \\ge 0$ and $x[k] = 0$ otherwise, with $a, b \\in \\C$.\nThe sum is a product of two geometric series, and $X(z_1, z_2) = \\frac{1}{(1 - a z_1^{-1})(1 - b z_2^{-1})}$ for $\\abs{z_1} \u003e \\abs{a}$ and $\\abs{z_2} \u003e \\abs{b}$.\n\nNon-example: $x[k] = 1$ for all $k \\in \\Z^2$ has an empty region of convergence.\nFor every $z$, choose signs $\\epsilon_1, \\epsilon_2 \\in \\set{1, -1}$ with $\\abs{z_j}^{-\\epsilon_j} \\ge 1$ for $j = 1, 2$.\nThen the infinitely many terms with $k_j = \\epsilon_j m_j$, $m_1, m_2 \\ge 0$, all have absolute value at least $1$, so the series diverges.\n",
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