Analytic continuation of multiple polylogarithms
Using the theory of Chen iterated path integrals, the author studies the problem of analytic continuation of the multiple polylogarithms \[ \operatorname{Li}_{m_1,\dots,m_n}(x_1,\dots,x_n):=\sum_{0<k_1<\dots<k_n}\frac{x_1^{k_1}\cdots x_n^{k_n}}{k_1^{m_1}\cdots k_n^{m_n}},\quad | x_j| <1,\; m_j\in\mathbb N,\; j=1,\dots,n. \] In particular, he proves that \(\operatorname{Li}_{m_1,\dots,m_n}\) extends to a multi-valued holomorphic function in \[ S_n:=\Big\{(x_1,\dots,x_n)\in\mathbb C^n: \prod_{1\leq j\leq n}(1-x_j) \prod_{1\leq j<k\leq n}(1-x_j\cdots x_k)\neq0\Big\}. \]
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