The effect of the presence of singularities on the method of stationary phase for a certain integral of two variables is examined. The singularities lie on two straight lines whose positions depend upon parameters. For certain values of the parameters the straight lines can pass through the point of stationary phase.
The asymptotic development, which is uniform with respect to the parameters, is determined. It is found that it can be expressed in terms of a function which, in certain circumstances, reduces to a Fresnel integral. The main properties of this function are derived in an appendix.
Explicit formulae are given for the dominant asymptotic terms in the cases when the point of stationary phase is approached by (i) neither line, (ii) only one of the lines, (iii) both lines.