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

Brian Fisher and Adem Kılıçman

Journal of Applied Mathematics, 2011, vol. 2011, 1-13

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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:846736

DOI: 10.1155/2011/846736

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