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Expansions of multiple integrals w(aixi,.. .,anxn)g(xi,... ,x")dxi ■ ■ -dxn, r°i r°n I " Ja\ J a" where w is a function on R" which is (bk -oj,)-periodic in the fcth variable, k = 1,... ,n, and g is smooth, are given in terms of negative powers of the integers o\,.. .,an. Estimates of the remainder term in the expansion are also given.doi:10.1090/s0025-5718-1988-0917829-6 fatcat:mtmnowz6q5cl3cqkvjzlohpumy