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On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions
Joint Authors
Source
Journal of Applied Mathematics
Issue
Vol. 2011, Issue 2011 (31 Dec. 2011), pp.1-13, 13 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2011-08-01
Country of Publication
Egypt
No. of Pages
13
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.
It is proved that the neutrix composition δ(rs-1)((tanhx+)1/r) exists and δ(rs-1)((tanhx+)1/r)=∑k=0s-1∑i=0Kk((-1)kcs-2i-1,k(rs)!/2sk!)δ(k)(x) for r,s=1,2,…, where Kk is the integer part of (s-k-1)/2 and the constants cj,k are defined by the expansion (tanh-1x)k={∑i=0∞(x2i+1/(2i+1))}k=∑j=k∞cj,kxj, for k=0,1,2,….
Further results are also proved.
American Psychological Association (APA)
Fisher, Brian& Kiliçman, Adem. 2011. On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions. Journal of Applied Mathematics،Vol. 2011, no. 2011, pp.1-13.
https://search.emarefa.net/detail/BIM-502917
Modern Language Association (MLA)
Fisher, Brian& Kiliçman, Adem. On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions. Journal of Applied Mathematics No. 2011 (2011), pp.1-13.
https://search.emarefa.net/detail/BIM-502917
American Medical Association (AMA)
Fisher, Brian& Kiliçman, Adem. On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions. Journal of Applied Mathematics. 2011. Vol. 2011, no. 2011, pp.1-13.
https://search.emarefa.net/detail/BIM-502917
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-502917