Contribution to the General Linear Conjugation Problem for A Piecewise Analytic Vector

Abstract

Establishing an analogy between the theories of Riemann–Hilbert vector problem and linear ODEs, for the n-dimensional homogeneous linear conjugation problem on a simple smooth closed contour Γ partitioning the complex plane into two domains D⁺ and D⁻ we show that if we know n−1 particular solutions such that the determinant of the size n−1 matrix of their components omitting those with index k is nonvanishing on D⁺ ∪ Γ and the determinant of the matrix of their components omitting those with index j is nonvanishing on Γ ∪ D⁻ {∞}, where \(k,j = \overline {1,n} \), then the canonical system of solutions to the linear conjugation problem can be constructed in closed form.