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(r), where F-n(x) = F(x) * delta(n)(x) for n = 1, 2, ... and {delta(n)(x)} is a certain regular sequenceconverging to the Dirac delta function. It is proved that the neutrix composition delta(s)[cosh(-1) (x(+)(1/r) +1)] exists and
delta(s)[cosh(-1) (x(+)(1/r) +1) - - Sigma(M-1)(k=0) Sigma(kr+r)(i=0) ((k)(i)) (-1)(i+k) rc(r,s,k)/(kr + r)k! delta((k)) (x),
for s = M - 1, M, M + 1, ... and r = 1,2,..., where
cr,s,k = Sigma(i)(j=0)((i)(j)) = (-1)(kr + r) (- i)/((2j - i)s + 1)/2(s + i + 1)
M is the smallest integer for which s - 2r + 1 < 2 Mr and r <= s/(2M + 2).
Further results are also proved.