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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 $0\leq\arg X\leq\pi$ and $Y$ is real is as follows: $$P'(X,Y)=\int\sb{-\infty}\sp \infty\exp[i(u\sp 4+Xu\sp 2+Yu)]du.$$ 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 $(\pi/8)$ and then defines the new variables $$t=ue\sp{-i\pi/8},\qquad x=Xe\sp{-i\pi/4},\qquad y=Ye\sp{i\pi/4},$$ which reduces $P'(X,Y)$ to $$P(x,y)=2e\sp{\pi i/8}\int\sb 0\sp \infty\exp(-t\sp 4-xt\sp 2)\cos(yt)dt$$ or, by introducing the Weber function also, to $$P(x,y)=2\sp{-1/4}\pi\sp{1/2}e\sp{(x\sp 2+i\pi)/8}{1 \over 2\pi i}\int\sb C {{\Gamma\sb{(s)}D\sb{s-1/2}({x\over\sqrt 2})}\over {(y\sp 2/4\sqrt 2 )\sp s}}ds\quad (y\ne0).$$ 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 $\vert X\vert\to\infty$, $Y$ finite. In the case of $\vert Y\vert\to\infty$, $X$ finite only the first terms in the expansion are given. The asymptotic behaviour on the caustic $Y\sp 2+(2X/3)\sp 3=0$ is also obtained.

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