The Riemann-Hilbert Problem and Fuchsian Differential Equations on the Riemann Sphere

A. A. Bolibruch
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A. A. Bolibruch: Steklov Mathematical Institute

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 1159-1168 from Springer

Abstract: Abstract (1) The Riemann-Hilbert problem concerns a certain class of linear ordinary differential equations (ODEs) in the complex domain. Let the system (1) $$ \frac{{dy}}{{dx}} = B(x)y$$ with unknown vector function y = (y1,…,y p )t (t means transposition) have singularities a1,…,an; that is, B(x) is holomorphic in S := $$\bar{\mathbb{C}} $$ \ {a1,…,an} (where $$\bar{\mathbb{C}} $$ is the Riemann sphere). The system is called Fuchsian at a i (and i is a Fuchsian singularity of the system) if B(x) has a pole there of order at most one. The system is Fuchsian if it is Fuchsian at all a i . Let all a i ≠ ∞. Then (2) $$ B(x) = \mathop \sum \limits_{i = 1}^n \frac{1}{{x - {a_i}}}{B_i},\mathop \sum \limits_{i = 1}^n {B_i} = 0.$$ .

Date: 1995
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DOI: 10.1007/978-3-0348-9078-6_109

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