On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions

摘要

Let F be a distribution in O' 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(x), where F_n(x) = F(x) * δ_n(x) for n = 1,2, ... and {δ_n (x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ(rs-1)( (tanh X+)1/r ) exists and δ~(rs-1)((tanh x+)~(1/r) = Σ_(k=0)~(s-1)((-1)~k cs~(-2),-1,k(r s)!/2sk!)δ(k) (x) for r, s = 1,2, ..., where K_k is the integer part of (s — k — 1)/2 and the constants c_(j,k) are defined by the expansion (tanh~(-1)x)~k={Σ_(i=0)~∞(x~(2i+1)/(21+1))}~k1)) = Σ_(j=k)~∞ c_(j,k) x~j, for k = 0,1,2, .... Further results are also proved.