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On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions
Joint Authors
Source
Journal of Applied Mathematics
Issue
Vol. 2011, Issue 2011 (31 Dec. 2011), pp.1-12, 12 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2011-06-01
Country of Publication
Egypt
No. of Pages
12
Main Subjects
Abstract EN
Let F be a distribution in D' and let f be a locally summable function.
The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(x)} is a certain regular sequence converging to the Dirac delta function.
In the ordinary sense, the composition δ(s)[(sinh-1x+)r] does not exists.
In this study, it is proved that the neutrix composition δ(s)[(sinh-1x+)r] exists and is given by δ(s)[(sinh-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), for s=0,1,2,… and r=1,2,…, where cs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!).
Further results are also proved.
American Psychological Association (APA)
Fisher, Brian& Kiliçman, Adem. 2011. On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions. Journal of Applied Mathematics،Vol. 2011, no. 2011, pp.1-12.
https://search.emarefa.net/detail/BIM-485043
Modern Language Association (MLA)
Fisher, Brian& Kiliçman, Adem. On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions. Journal of Applied Mathematics No. 2011 (2011), pp.1-12.
https://search.emarefa.net/detail/BIM-485043
American Medical Association (AMA)
Fisher, Brian& Kiliçman, Adem. On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions. Journal of Applied Mathematics. 2011. Vol. 2011, no. 2011, pp.1-12.
https://search.emarefa.net/detail/BIM-485043
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-485043