Journal of Applied Mathematics

On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions

Brian Fisher and Adem Kılıçman

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Abstract

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 { F n ( f ( x ) ) } is equal to h ( x ) , where F n ( x ) = F ( x ) * δ n ( x ) for n = 1,2 , and { δ n ( x ) } is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ ( r s - 1 ) ( ( tanh x + ) 1 / r ) exists and δ ( r s - 1 ) ( ( tanh x + ) 1 / r ) = k = 0 s - 1 i = 0 K k ( ( - 1 ) k c s - 2 i - 1 , k ( r s ) ! / 2 s k ! ) δ ( k ) ( x ) for r , s = 1,2 , , where K k is the integer part of ( s - k - 1 ) / 2 and the constants c j , k are defined by the expansion ( tanh - 1 x ) k = { i = 0 ( x 2 i + 1 / ( 2 i + 1 ) ) } k = j = k c j , k x j , for k = 0 , 1 , 2 , . Further results are also proved.

Article information

Source
J. Appl. Math., Volume 2011 (2011), Article ID 846736, 13 pages.

Dates
First available in Project Euclid: 15 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1331818616

Digital Object Identifier
doi:10.1155/2011/846736

Mathematical Reviews number (MathSciNet)
MR2820079

Zentralblatt MATH identifier
1236.46035

Citation

Fisher, Brian; Kılıçman, Adem. On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions. J. Appl. Math. 2011 (2011), Article ID 846736, 13 pages. doi:10.1155/2011/846736. https://projecteuclid.org/euclid.jam/1331818616


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