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Published in:
2020 | OriginalPaper | Chapter
11. Linear Pfaffian Systems and Integrability Condition
Author : Yoshishige Haraoka
Published in: Linear Differential Equations in the Complex Domain
Publisher: Springer International Publishing
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Abstract
We denote by x = (x
1, x
2, …, x
n) the coordinate of \(\mathbb {C}^n\). Let N be an integer. Let \(X\subset \mathbb {C}^n\) be a domain, and \(a_{ij}^k(x)\) (1 ≤ i, j ≤ N, 1 ≤ k ≤ n) be holomorphic functions on X. The system of partial differential equations of the first order with unknown functions u
1(x), u
2(x), …, u
N(x) is called a linear Pfaffian system.
$$\displaystyle \frac {\partial u_i}{\partial x_k}=\sum _{j=1}^Na_{ij}^k(x)u_j \quad (1\leq i\leq N,\,1\leq k\leq n) $$