# V.P. Krivokolesko. On computing a class of integrals of rational functions with parameters and singularities on complex hyperplanes ... P. 123-140

We give an algorithm for computing the integral
$$\displaystyle\int_{|\xi_1|=1}\ldots\displaystyle\int_{|\xi_n|=1}\frac{f(\xi)}{ \prod \limits_{j=1}^m (a_{j,1}z_1 \xi_1+\ldots+a_{j,n}z_n \xi_n+c_j)^{t_j}}\cdot \frac{d\xi_1}{\xi_1}\ldots\frac{d\xi_n}{\xi_n},$$
where the integration set is the distinguished boundary of the unit polydisk in $\mathbb C^n$, the function $f(\xi)$ is holomorphic in a neighborhood of this set, and $\prod_{j=1}^m (a_{j,1}z_1 \xi_1+\ldots+a_{j,n}z_n \xi_n+c_j)\not=0$ for points $z=(z_1,\ldots, z_n)$ of a connected $n$-circular set $G\subset\mathbb C^n$. For points of the distinguished boundary, whose coordinates satisfy the relations $|\xi_1|=1$, $\ldots$, $|\xi_n|=1$, the sets $\{V_j\}=\{(z_1,\ldots,z_n)\in\mathbb C^n\colon a_{j,1}z_1 \xi_1+\ldots+a_{j,n}z_n \xi_n+c_j=0\}$ are $n$-circular, and it is convenient to study their mutual arrangement in $\mathbb C^n$ by using the projection $\pi\colon \mathbb C^n\rightarrow \mathbb R^n_{+}$, where $\pi(z_1,\ldots,z_n)=(|z_1|,\ldots,|z_n|)$. A connected set $\pi(\{V_j\})$ divides $\mathbb R^n_+$ into at most $n+1$ disjoint nonempty parts, and $\pi(G)$ belongs to one of them. Therefore the number of variants of the mutual arrangement of the sets $G$ and $\{V_1\},\ldots,\{V_m\}$ in $\mathbb C^n$, which influences the value of the integral, does not exceed $(n+1)^m$. In Theorems 1 and 2 we compute the integral for two of these variants. An example of computing a double integral by applying its parameterization and one of the theorems is given.

Keywords: integral representation, $n$-circular domain, complex plane.

The paper was received by the Editorial Office on October 9, 2017.

Funding Agency:

1) Ministry of Education and Science of the Russian Federation (Grant Number НШ-9149.2016.1);

2) grant from the government of the Russian Federation to conduct research under the guidance of leading scientists at the Siberian Federal University (project no. 14.Y26.31.0006).

Viacheslav Pavlovich Krivokolesko, Cand. Sci. ( Phys.-Math.), Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: krivokolesko@gmail.com.

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ISBN: 978-0-8218-4560-8 .