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

V. P. Krivokolesko

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 123-140

Résumé

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.

Détail

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