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

Abstract

LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix compositionδ(rs-1)((tanhx+)1/r)exists andδ(rs-1)((tanhx+)1/r)=∑k=0s-1∑i=0Kk((-1)kcs-2i-1,k(rs)!/2sk!)δ(k)(x)forr,s=1,2,…, whereKkis the integer part of(s-k-1)/2and the constantscj,kare defined by the expansion(tanh-1x)k={∑i=0∞(x2i+1/(2i+1))}k=∑j=k∞cj,kxj, fork=0,1,2,….Further results are also proved.