On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse 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. It is proved that the neutrix composition δ rs−1 tanhx 1/r exists and δ rs−1 tanhx 1/r ∑s−1 k 0 ∑Kk i 0 −1 cs−2i−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 cj,k are defined by the