- 期刊
ON DEFINING THE GENERALIZED FUNCTIONS δ^α(z) AND δ^n(x)
摘要
In a previous paper (see [5]), we applied a fixed δ-sequence and neutrix limit due to Van der Corput to give meaning to distributions δ^k and (δ')^k for k ∈ (0, 1) and k = 2, 3, .... In this paper, we choose a fixed analytic branch such that z^α(-π ＜ arg z ≤ π) is an analytic single-valued function and define δ^α(z) on a suitable function space I_a. We show that δ^α(z) ∈ I'_a. Similar results on (δ^((m))(z))^α are obtained. Finally, we use the Hilbert integral φ(z) = 1/πi ∫_(-∞)^(+∞) φ(t)/t-z dt where φ(t) ∈ D(R), to redefine δ^n(x) as a boundary value of δ^n(z-iϵ). The definition of δ^n(x) is independent of the choice of δ-sequence.
延伸閱讀
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