Citation
Fisher, Brian and Kilicman, Adem
(2011)
On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions.
Journal of Applied Mathematics, 2011 (846736).
pp. 114.
ISSN 1110757X; ESSN:16870042
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 {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(rs1) ((tanh x+)1/r) exists and δ(rs1) ((tanh x+)1/r) =√k=0 s1√i=0 Kk (( 1)k cs2 i  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 c j,k are defined by the expansion (tanh  1 x)k = {√i=0 ∞ (x 2i+1/(2 i + 1)) }k = √j=k ∞ c j, k xj, for k = 0,1, 2,⋯. Further results are also proved.
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