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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 2, Pages 123–140 (Mi timm1528)

On computing a class of integrals of rational functions with parameters and singularities on complex hyperplanes

V. P. Krivokolesko

Siberian Federal University, Krasnoyarsk

Abstract: 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.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÍØ-9149.2016.114.Y26.31.0006

DOI: https://doi.org/10.21538/0134-4889-2018-24-2-123-140

Full text: PDF file (269 kB)
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Bibliographic databases:

UDC: 517.55+519.117
MSC: 32A07, 32A26, 05A19

Citation: V. P. Krivokolesko, “On computing a class of integrals of rational functions with parameters and singularities on complex hyperplanes”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 2, 2018, 123–140

Citation in format AMSBIB
\Bibitem{Kri18} \by V.~P.~Krivokolesko \paper On computing a class of integrals of rational functions with parameters and singularities on complex hyperplanes \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2018 \vol 24 \issue 2 \pages 123--140 \mathnet{http://mi.mathnet.ru/timm1528} \crossref{https://doi.org/10.21538/0134-4889-2018-24-2-123-140} \elib{https://elibrary.ru/item.asp?id=35060683}