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Saddle-point method and resurgent analysis. (English. Russian original) Zbl 0915.58067
Math. Notes 61, No. 2, 227-241 (1997); translation from Mat. Zametki 61, No. 2, 278-296 (1997).
Summary: The topological part of the theory of the parameter-dependent Laplace integral is known to consist of two stages. At the first stage, the integration contour is reduced to a sum of paths of steepest descent for some value of the parameter. At the second stage, this decomposition (and hence the asymptotic expansion of the integral) is continued to all other parameter values. In the present paper, the second stage is studied with the help of resurgent analysis techniques.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
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References:
[1] J. Écalle,Les fonctions résurgentes. I, II, III, Publ. Mathématiques d’Orsay, Paris (1981–1985).
[2] B. Candelpergher, J. C. Nosmas, and F. Pham,Approche de la résurgence, Hermann éditeurs des sciences et des arts, Paris (1993).
[3] B. Sternin and V. Shatalov, ”On a notion of resurgent function of several variables,”Math. Nachr.,171, 283–301 (1995). · Zbl 0842.32006
[4] B. Sternin and V. Shatalov,Borel-Laplace Transform and Asymptotic Theory, CRC Press, Florida (1995). · Zbl 0852.34001
[5] M. V. Berry and C. J. Howls, ”Hyperasymptotics for integrals with saddles,”Proc. Roy. Soc. London. Ser. A.,443, 657–675 (1991). · Zbl 0764.30031
[6] B. Malgrange, ”Méthode de la phase stationnaire et sommation de Borel,” in:Complex Analysis. Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys, Vol. 126, Springer Verlag, Berlin-Heidelberg (1980).
[7] F. Pham, ”Vanishing homologies and then variable saddlepoint method,” in:Proc. Sympos. Pure Math, Vol. 40, Part 2, AMS, Providence (1983), pp. 319–333.
[8] B. Sternin and V. Shatalov,Saddle Point Method and Resurgent Analysis, Preprint No. 95-69, Max-Planck-Institut für Mathematik, Bonn (1995). · Zbl 0915.58067
[9] B. Sternin and V. Shatalov,Differential Equations on Complex Manifolds, Kluwer Acad. Publ., Dordrecht (1994). · Zbl 0818.35003
[10] M. V. Fedoryuk,The Saddle-Point Method [in Russian], [Russian translation], Nauka, Moscow (1977). · Zbl 0463.41020
[11] J. C. Tougeron,An Introduction to the Theory of Gevrey Expansions and to Borel-Laplace Transform, with Some Applications, Preprint, Univ. of Toronto, Canada (1990).
[12] S. Lefschetz,L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris (1924).
[13] E. Delabaere, ”Introduction to the Écalle theory,” in:Computer Algebra and Differential Equations London Math. Soc. Lecture Note Ser, Vol. 193 (E. Tournier, editor), Univ. Press, Cambridge (1994), pp. 59–102. · Zbl 0805.40007
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