Integrals are considered of the form
is a contour in the complex plane with one finite endpoint. Earlier work of M. V. Berry
and C. J. Howls
[Proc. R. Soc. Lond., Ser. A 434, No. 1892, 657-675 (1991; Zbl 0764.30031
)] is continued in studying exponentially accurate asymptotics of this class of integrals; in Berry and Howls (1991) contours with no finite endpoints were considered. The new results are illustrated by application to the complementary error function and an incomplete Airy function.