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On a formula for the asymptotic expansion of an integral in complex analysis. (English. Russian original) Zbl 0753.30027
Sov. Math., Dokl. 43, No. 2, 624-627 (1991); translation from Dokl. Akad. Nauk SSSR 317, No. 6, 1311-1314 (1991).
The stationary phase method is developed for the case of multivalued analytic functions depending on a parameter. Applications in the theory of differential equations are in view.
Let $$X\subset\mathbb{C}^ n$$ be an algebraic subset of codimension 1, $$\tilde x\in X$$ be a point such that the tangent plane at $$\tilde x$$ to $$X$$ has a quadratic contact, $$h(p)$$ be the $$(n-1)$$-dimensional vanishing homology class of the pair $$(L(p_ 0,p)$$, $$L(p_ 0,p)\cap X)$$ where $$L(p_ 0,p)=\{x;p_ 0+px=0\}$$ is close to the tangent plane. The asymptotic expansion is given for the Radon integral $Rf=\int_{h(p)}\text{Res}{f(x)dx\over p_ 0+px}$ for a function $$f\in{\mathcal {AF}}_{\alpha,q}(X)$$. Here $${\mathcal {AF}}_{\alpha,q}$$ are multivalued analytic functions $$f$$ such that $$f(x)=s(x)^ \alpha g(x)$$; $$s(x)=0$$ is the equation of $$X$$, $$g$$ satisfies $$| D^ \alpha g|\leq C$$ if $$|\alpha|\leq M-1$$, $$| D^ \alpha g|\leq C| s|^{q-n}$$ if $$|\alpha|=M$$ where $$M=q-q'$$ with $$- 1<q'<0$$. The operator $$R$$ acts between the spaces $${\mathcal {AF}}_{\alpha,q}$$ and the asymptotic expansion consists (roughly saying) in determination of the induced mapping between the factors $${\mathcal J}^{M,\alpha}={\mathcal {AF}}_{\alpha,q}/{\mathcal A}_{q+\alpha}$$ where $${\mathcal A}_ r$$ are the classes of multivalued analytic functions $$f$$ that satisfy $$| f(x)|\leq C| s(x)|^ r$$.
Reviewer: J.Chrastina (Brno)

##### MSC:
 30E15 Asymptotic representations in the complex plane 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)