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.
REFERENCES
1. Aizenberg L.A., Yuzhakov A.P. Integral representations and residues in multidimensional complex analysis. Providence, AMS, 1983, 283 p. ISBN: 978-0-8218-1550-2 . Original Russian text published in Aizenberg L.A., Yuzhakov A.P. Integral’nye predstavleniya i vychety v mnogomernom kompleksnom analize, Novosibirsk, Nauka Publ. (Sibirsk. Otdel.), 1979, 366 p.
2. Egorychev G.P. Integral representation and the computation of combinatorial sums. Transl. Math. Monographs, vol. 59. Providence, AMS, 1984, 286 p. ISBN: 0821845128 . Original Russian text published in Egorychev G.P. Integral’noe predstavlenie i vychislenie kombinatornykh summ. Nauka Publ. (Sibirsk. Otdel.), 1977, 271 p.
3. Krivokolesko V.P. Method for obtaining combinatorial identities with polynomial coefficients by the means of integral representations. J. Sib. Fed. Univ. Math. Phys., 2016, vol. 9, no. 2, pp. 192–201. doi: 10.17516/1997-1397-2016-9-2-192-201 .
4. Pham F. Introduction $\grave{a}$ l’etude topologique des singularites de Landau. [Introduction a l’etude topologique des singularites de Landau]. Paris, Gauthier Villars, 1967, 141 p. ISBN: 286883762X . Translated to Russian under the title Vvedenie v topologicheskoe issledovanie osobennostei Landau, Moscow, Mir Publ., 1967, 184 p.
5. Hwa R., Teplitz V. Homology and Feynman Integrals. [Mathematical Physics Monograph Series]. New York, Amsterdam, Benjamin, 1966, 331 p. ISBN: 9780805347500 . Translated to Russian under the title Gomologii i feinmanovskie integraly. Moscow, Mir Publ., 1969, 229 p.
6. Krivokolesko V.P. Integral representations for linearly convex polyhedra and some combinatorial identities. J. Sib. Fed. Univ. Math. Phys., 2009, vol. 2, no. 2, pp. 176–188 (in Russian).
7. Krivokolesko V.P., Tsikh A.K. Integral representations in linearly convex polyhedra. Sib. Math. J., 2005, vol. 46, no. 3, pp. 453–466. doi: 10.1007/s11202-005-0048-4 .
8. Forsberg M., Passare M., Tsikh A. Laurent determinants and arrangements of hyperplane amoebas. Advances in Math., 2000, vol. 151, no. 1, pp. 45–70. doi: 10.1006/aima.1999.1856 .
9. Shabat B.V. Introduction to complex analysis. Part II: Functions of several variables. Providence, AMS, 1992, 371 p. ISBN: 082189739X . Original Russian text (Parts I,II, 1st ed.) published in Vvedenie v kompleksnyi analiz, Moscow, Nauka Publ., 1969, 576 p.
10. Tsikh A.K. Multidimensional residues and their applications. Providence, AMS, 1992, 188 p.
ISBN: 978-0-8218-4560-8 .
