Global asymptotics for multiple integrals with boundaries.
The authors extend results given by \textit{C. J. Howls} [Proc. R. Soc. Lond., Ser. A 453, No. 1966, 2271--2294 (1997; Zbl 1067.58501)] for integrals of the following type: \[ I_\Gamma(k)= \int_\Gamma e^{-kf(z)}g(z)\,dz^{(1)}\wedge \cdots\wedge dz^{(n)}, \] where \(f,g:{\mathbb C}\to{\mathbb C}\) are analytic functions in \(z=(z^{(1)},\dots,z^{(n)})\) and \(\Gamma\) is a chain of real dimension \(n\) with \(m\leq n\) smooth irreducible affine hypersurfaces as boundaries. Based on a geometric decomposition of the chain \(\Gamma\) [cf. \textit{F. Pham}, Astérisque 130, 11--47 (1985; Zbl 0597.32012)] the asymptotic expansion follows from a local analysis near the critical points. In a well accessible manner the authors study the asymptotic expansion from a geometrical and an analytical point of view. They analyse Stokes phenomenon (resp. Picard-Lefschetz formulae) and calculate exact remainder terms for truncated asymptotic expansions. Further on, Stokes constants and hyperasymptotics are given. An example illustrates the derived techniques.
scientific article; zbMATH DE number 1820889
scientific article