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  "body": "\n## Summary\n\nThe multiple gamma functions $\\Gamma_N(w \\mid a_1, \\ldots, a_N)$ of E. W. Barnes generalise the [[gamma function|0H5SYBSED9]] to $N$ periods.\nThey are built from the Barnes zeta function $\\sum (w + n_1 a_1 + \\cdots + n_N a_N)^{-s}$ by zeta regularisation, as the gamma function is built from the [[Hurwitz zeta function|G4SW7HJG88]].\nShifting $w$ by a period $a_j$ divides $\\Gamma_N$ by a multiple gamma function of order $N - 1$.\nThe [[Barnes G-function|7E2JJWFF00]] is essentially the case $N = 2$ with $a_1 = a_2 = 1$.\n\n## Explanations\n\n### L1 human\n\nThe factorial $n!$ multiplies the numbers $1, \\ldots, n$, which lie on a line with step $1$.\nOne can try to multiply all numbers $w + n_1 a_1 + n_2 a_2$ of a lattice in a quadrant, or in a higher-dimensional corner.\nThis infinite product diverges, but a standard regularisation trick gives it a finite meaning.\nThe result is the multiple gamma function.\nFor one period equal to $1$ it gives back $\\Gamma(w)$, up to the constant factor $1/\\sqrt{2\\pi}$.\n",
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