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Asymptotic expansion of a multiple integral. (English) Zbl 0636.41024
Die Verff. betrachten die asymptotische Entwicklung des Integrals $$ J(s)=\int\sp{1}\sb{0}\int\sp{1}\sb{0}g(x\sp ay\sp b/s)x\sp{\alpha}y\sp{\beta}f(x,y)dxdy $$ für $s\to +0$. Mittels Mellin Transformation wird J(s) durch das Integral $J(s)=(1/2\pi i)\int\sp{c+i\infty}\sb{c-i\infty}s\sp{-z}F(z)M[g,-z]dz$ dargestellt. Danach gibt die Verschiebung des Weges nach links die gesuchte Entwicklung. Die Hauptaufgabe ist die Eigenschaften des Integrals $F(z)=\int\sp{1}\sb{0}\int\sp{1}\sb{0}x\sp{\alpha +az}y\sp{\beta +bz}f(x,y)dxdy$ in der Z-Ebene zu untersuchen. Es wird gezeigt, daß F(z) eine meromorphe Funktion ist, und die Residuen werden berechnet.
Reviewer: E.Riekstinš
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
44A15Special transforms (Legendre, Hilbert, etc.)
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