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On the neutrix composition of the distributions \(x^{-s} \ln^m |x|\) and \(x^r\). (English) Zbl 1194.46061

Summary: Let \(F\) be a distribution and let \(f\) be a locally summable function. The distribution \(F(f)\) is defined as the neutrix limit of the sequence \(\{F_n (f)\}\), where \(F_n (x) = F(x) * \delta _n (x)\) and \(\{ \delta _n (x)\}\) is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function \(\delta (x)\). The composition of the distributions \(x^{-s} \ln^m |x|\) and \(x^r\) is proved to exist and be equal to \(r^m x^{-rs} \ln^m |x|\) for \(r,s,m = 2, 3\dots \).

MSC:

46F10 Operations with distributions and generalized functions
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References:

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