On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions - Brian Fisher; Adem Kılıçman (Journal of Applied Mathematics)
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)=∑k=0s-1∑i=0Kk((-1)kcs-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 expansion (tanh-1x)k={∑i=0∞(x2i+1/(2i+1))}k=∑j=k∞cj,kxj, for k=0,1,2,…. Further results are also proved.