Asymptotics of functions that are a generalization of the euler gamma function

The asymptotic behavior of two classes of functions expressed by integrals is considered in the paper. For special values of the parameters the functions of these classes are 1/(z) or (z+1). The necessity of their study is occasioned by the connection problem for an ordinary linear differential equation with two singular points. By means of a number of lemmas and the method of descent, a theorem is proved which gives the asymptotics of these functions as ¦z¦ in some sector.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 123–129, 1981.