The central connection problem at turning points of linear differential equations

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Abstract
A system of linear differential equations of the vectorial form εdy/dx=A (x, ε) y is considered, where ε is a positive parameter, and the matrixA (x, ε) is holomorphic in |x|⩽x 0, 0 < ε ⩽ ε0 , with an asymptotic expansionsA (x, ε) ∼ ∑ ∞ r=0 A r (x) εr, as ε→0. The eigenvalues ofA 0(x) are supposed to coalesce atx=0 so as to make this point a simple turning point. With the help of refinements of the representations for the inner and outer asymptotic solutions, as ε→0, that were introduced in the articles [9] and [10] by the author (see the references at the end of the paper), explicit connection formulas between these solutions are calculated. As part of this derivation it is shown that only the diagonal entries of the connection matrix are asymptotically relevant.