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  "body": "\n## Definition\n\nLet $N \\ge 1$ be an integer, let $a = (a_1, \\ldots, a_N)$ with real $a_i \u003e 0$, and let $w$ be a complex number with $\\operatorname{Re} w \u003e 0$.\nThe *Barnes zeta function* is\n$$\\zeta_N(s, w \\mid a) = \\sum_{n_1, \\ldots, n_N \\ge 0} \\frac{1}{(w + n_1 a_1 + \\cdots + n_N a_N)^s}, \\qquad \\operatorname{Re} s \u003e N,$$\nwith the principal branch of the power.\nIt has a meromorphic [[analytic continuation|0M3YQJ81NB]] in $s$ to $\\C$ that is holomorphic at $s = 0$.\nThe *multiple gamma function* of order $N$ is\n$$\\Gamma_N(w \\mid a) = \\exp\\left(\\left.\\frac{\\partial}{\\partial s} \\zeta_N(s, w \\mid a)\\right|_{s = 0}\\right).$$\n\n## Remarks\n\nExample: for $N = 1$ and $a_1 = 1$, $\\zeta_1(s, w \\mid 1)$ is the [[Hurwitz zeta function|G4SW7HJG88]] $\\zeta(s, w)$.\nLerch's formula $\\frac{\\partial}{\\partial s}\\zeta(s, w)\\big|_{s=0} = \\log \\Gamma(w) - \\frac12 \\log 2\\pi$ gives $\\Gamma_1(w \\mid 1) = \\Gamma(w)/\\sqrt{2\\pi}$, where $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\n\nThe identity $\\zeta_N(s, w \\mid a) - \\zeta_N(s, w + a_j \\mid a) = \\zeta_{N-1}(s, w \\mid a^{(j)})$, where $a^{(j)}$ omits $a_j$, gives the functional equation\n$$\\Gamma_N(w + a_j \\mid a) = \\frac{\\Gamma_N(w \\mid a)}{\\Gamma_{N-1}(w \\mid a^{(j)})},$$\nwith the convention $\\Gamma_0(w) = 1/w$.\nFor $N = 1$ it reads $\\Gamma_1(w + 1 \\mid 1) = w\\, \\Gamma_1(w \\mid 1)$, the functional equation of $\\Gamma$.\n\nNon-example of a valid parameter: if $a_1 = 0$, each term is repeated for every value of $n_1$, the series diverges for every $s$, and $\\Gamma_N$ is not defined.\n\nSeveral normalisations are in use; they differ by factors of the form $\\exp(\\text{polynomial in } w)$.\nThe function $1/\\Gamma_N(w \\mid a)$ extends to an entire function of $w$.\n",
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