On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions
Summary: Let \(F\) be a distribution in \(\mathcal D'\) and let \(f\) be a locally summable function. The composition \(F(f(x))\) of \(F\) and \(f\) is said to exist and to be equal to the distribution \(h(x)\) if the limit of the sequence \(\{F_n(f(x))\}\) is equal to \(h(x)\), where \(F_n(x) = F(x) \ast {\delta}_n(x)\) for \(n = 1, 2, \dots\) and \(\{{\delta}_n(x)\}\) is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition \({\delta}^{(rs-1)}((\tanh x_+)^{1/r})\) exists and \({\delta}^{(rs-1)}((\tanh x_+)^{1/r} = \sum^{s-1}_{k=0} \sum^{K_k}_{i=0} ((-1)^k c_{s-2i-1,k}(rs)!/2sk!)\delta^{(k)}(x)\) for \(r, s = 1, 2, \dots\), where \(K_k\) is the integer part of \((s - k - 1)/2\) and the constants \(c_{j,k}\) are defined by the expansion \(({\tanh}^{-1}x)^k = \{\sum^{\infty}_{i=0} (x^{2i+1}/(2i + 1))\}^k = \sum^{\infty}_{j=k} c_{j,k} x^{j}\), for \(k = 0, 1, 2, \dots\) . Further results are also proved.
scientific article; zbMATH DE number 5964460
scientific article