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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)

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