The aim of the author is to investigate the asymptotic behaviour of the function , a representation of which when and is real is as follows:
In order to get the analytic continuation of for general complex values of and the author first rotates the path of integration through an angle of and then defines the new variables
which reduces to
or, by introducing the Weber function also, to
This last integral is absolutely convergent for all complex values of and . By using this last representation he derives a full asymptotic expansion of for the case , finite. In the case of , finite only the first terms in the expansion are given. The asymptotic behaviour on the caustic is also obtained.