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The asymptotic behaviour of Pearcey’s integral for complex variables. (English) Zbl 0739.33014

The aim of the author is to investigate the asymptotic behaviour of the function P ' (X,Y), a representation of which when 0argXπ and Y is real is as follows:

P ' (X,Y)= - exp[i(u 4 +Xu 2 +Yu)]du·

In order to get the analytic continuation of P ' (X,Y) for general complex values of X and Y the author first rotates the path of integration through an angle of (π/8) and then defines the new variables

t=ue -iπ/8 ,x=Xe -iπ/4 ,y=Ye iπ/4 ,

which reduces P ' (X,Y) to

P(x,y)=2e πi/8 0 exp(-t 4 -xt 2 )cos(yt)dt

or, by introducing the Weber function also, to

P(x,y)=2 -1/4 π 1/2 e (x 2 +iπ)/8 1 2πi C Γ (s) D s-1/2 (x 2) (y 2 /42) s ds(y0)·

This last integral is absolutely convergent for all complex values of x and y. By using this last representation he derives a full asymptotic expansion of P ' (X,Y) for the case |X|, Y finite. In the case of |Y|, X finite only the first terms in the expansion are given. The asymptotic behaviour on the caustic Y 2 +(2X/3) 3 =0 is also obtained.


MSC:
33E20Functions defined by series and integrals
30E15Asymptotic representations in the complex domain
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)