Asymptotics of some functions generalizing the Euler gamma-function

Asymptotic behavior of two classes of functions defined by some integrals is considered. The functions $1/\Gamma(z)$ and $1/\Gamma(z+1)$ are examples of functions of this classes. The problem of investigation of this functions arises from the “connection problem” for a linear ordinary differential equations with two singular points. The theorem giving asymptotics of these functions when $|z|\to\infty$ in a certain sector is proved by making use of some lemmas and saddle point method.