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The neutrix composition  F ( f  ( x )) of a distribution  F ( x ) and a locally summable function  f  ( x ) is said to exist and be equal to the distribution  h ( x ) if the neutrix limit of the sequence { F n  ( f  ( x ))} is equal to  h ( x ), where  F n  ( x ) =  F ( x ) * δ  n  ( x ) and {δ  n  ( x )} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function ( x ). The functioncosh    +   -  1(   x  +  1   )$\cosh _ + ^{ - 1}\left( {x + 1} \right)$    is defined by

demonstratio mathematica

Further results on the neutrix composition of distributions involving the delta function and the function           cosh    +   -  1     (     x    1  /  r    +  1   )     $\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)$