On the Borel transform of the uniform asymptotic expansion of Bessel functions of large order (Q2744095)

In the present paper the author studies the Borel transform of the classical uniform asymptotic expansion (U.A.E.) of the Bessel function \(J_\nu (\nu z)\) [see \textit{M. Abramowitz} and \textit{I. A. Stegun}, Handbook of mathematical functions with formulas, graphs and mathematical tables (1964; Zbl 0171.38503)]. This Borel transform will be the function \(Y(t,\zeta)\) such that NEWLINE\[NEWLINEJ_\nu(\nu z)={1\over 2\pi i}\int_{-\infty}^{\Bigl(\pm{2\over 3}\zeta^{3/2}+\Bigr)}e^{\nu t}Y(t,\zeta)dtNEWLINE\]NEWLINE where the contour of integration starts at \(-\infty\), encircles the points \(\pm{2\over 3}\zeta^{3/2}\) in the positive sense and returns to the starting point. From the introduction: ``The Borel transform will be in terms of new functions that have coalescing branch points. These are also needed in Cauchy-type integral representations, involving the Borel transform, for the coefficients in the U.A.E. The other singularities of the Borel transform are located and the local expansions at these singularities are used to obtain asymptotic expansions for the late coefficients in the U.A.E. The fact that the nearest singularities are just copies of the singularities near the origin explains the property called resurgence''.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].