Asymptotics of a multiple Cauchy integral with rapidly oscillating exponential

An asymptotic expansion uniform with respect to a parameter is obtained for a multiple Cauchy integral with a rapidly oscillating exponential in the integrand under the assumption that the singularity lines are in a general position.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 231–242, August, 1995.The work was financially supported by the Russian Foundation for Basic Research under grant No. 94-01-00193-a.