On the neutrix composition of the delta and inverse hyperbolic sine functions

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. in the ordinary sense,the composition δ(s) [(sinh-1 x+)r] does not exists. in this study,it is proved that the neutrix composition δ(s) [ (sinh -1 x+)r ] exists and is given by δ(s) [ (sinh -1 x+)r]= ∑k=0 sr+r-1 ∑i=0 k (k i)((-1)k rc s,k,i /2k+1k!) δ(k) (x),for s = 0,1,2,⋯ and r = 1,2,⋯,where cs,k,i = (- 1)s s ! [(k - 2 i + 1)rs-1 + (k - 2i - 1)rs+r-1]/(2 (r s + r - 1) !). further results are also proved. © 2011 brian fisher and adem klman.