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# Asymptotic and transasymptotic matching; formation of sin- gularities

DOI link for Asymptotic and transasymptotic matching; formation of sin- gularities

Asymptotic and transasymptotic matching; formation of sin- gularities book

# Asymptotic and transasymptotic matching; formation of sin- gularities

DOI link for Asymptotic and transasymptotic matching; formation of sin- gularities

Asymptotic and transasymptotic matching; formation of sin- gularities book

## ABSTRACT

Transasymptotic matching stands for matching at the level of transseries. Matching can be exact, in that a BE summable transseries, valid in one region, is matched to another BE summable transseries, valid in an adjacent region, or asymptotic, when a transseries is matched to a classical asymptotic expansion. An example of exact matching is (5.122), with the connection formula (5.123), valid for systems of ODEs. In this case the two transseries exactly represent one function, and the process is very similar to analytic continuation; it is a process of continuation through transseries. The collection of matched transseries represents exactly the function on

the union of their domain of validity. In the case of linear ODEs, matching is global, in that it is valid in a full (ramified, since the solution might not be single-valued) neighborhood of infinity. In this case, by the results in Chapter 5 we see that for any φ we have

y = LφY+0 + ∑ |k|=1

Cke−k·λxxk·αLφY+k (6.1)

where we note that (due to linearity) the right side of (6.1) has finitely many terms. If φ corresponds to a Stokes ray, Lφ is understood as the balanced average. For linear systems, the Stokes rays are precisely the directions of the λi, i = 1, ..., n, that is, the directions along which xλi ∈ R+. Along the direction λi, the constant Ci in the transseries changes; see (5.123). Up to the changes in C, the representation (6.1) is uniform in a ramified neighborhood of infinity. We emphasized linear, since solutions of nonlinear ODEs usually develop

infinitely many singularities as we shall see, and even natural boundaries in a neighborhood of infinity, and in the latter case transasymptotic matching often ends there (though at times it suggests pseudo-analytic continuation formulas.) The information contained in the transseries suffices to determine, very accurately for large values of the variable, the position and often the type of singularities. We first look at a number of simple examples, which should provide the main ideas for a more general analysis, found in [24] together with rigorous proofs.