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Journals / Hacettepe Journal of Mathematics and Statistics / 2014 / Cilt: 43 - Sayı: 1

RESULTS ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION

Brian Fisheraand Biljana JOLEVSKATUNESKAB
Pages
43–50
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Further results are also proved.

📌 3 Key Takeaways

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Objective & Hypothesis:The neutrix composition F (f (x))) of a distribution F (x) and a locallysummable function f (x) is said to exist and be equal to the distributionh(x) if the neutrix limit of the sequence {Fn (f (x))} is equal to h(x),where Fn (x) = F (x) ∗ δn (x) and {δn (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x).
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Key Findings & Data:It is proved that the neutrix composition δ(s) {[exp + (x) − 1]r } exists andδ (s) {[exp + (x) − 1]r } = rs+r−1k=0 (−1)s+k s!c rs+r−1,k2rk! δ (k) (x), for r = 1, 2, . . . and s = 0, 1, 2, . . ..
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Conclusion & Impact:Further results are also proved.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:distribution · dirac-delta function · composition of distributions

Abstract

The neutrix composition F (f (x))) of a distribution F (x) and a locallysummable function f (x) is said to exist and be equal to the distributionh(x) if the neutrix limit of the sequence {Fn (f (x))} is equal to h(x),where Fn (x) = F (x) ∗ δn (x) and {δn (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδ(x). It is proved that the neutrix composition δ(s) {[exp + (x) − 1]r } exists andδ (s) {[exp + (x) − 1]r } = rs+r−1k=0 (−1)s+k s!c rs+r−1,k2rk! δ (k) (x), for r = 1, 2, . . . and s = 0, 1, 2, . . .. Further results are also proved.

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Cite This Article

JOLEVSKATUNESKAB, B. F. B. (2022). RESULTS ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION. Hacettepe Journal of Mathematics and Statistics, Cilt: 43 - Sayı: 1, 43–50. https://ajindex.com/articles/530391

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References (10)

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