Hyperfunctions Depending on Parameters

Abstract

In the previous section, as specific examples of hyperfunctions, we explained hyperfunctions of the form x ^-m , x ^-m H(x), x ^-m sgn x and (log ∣x∣)ⁿ (m, n positive integers). The concept of a formal product, i.e. hyperfunctions of the form ψ(x)f(x) with a hyperfunction f(x) and a single-valued analytic function ψ(x), played a basic role. Moreover, hyperfunctions ∣x∣^α, ∣x∣^αH(x), ∣x∣^α sgn x etc. were defined for α complex. What are the relations between them and x ^-m , x ^-m H(x) etc? To investigate this, we now discuss hyperfunctions f(x,α) depending on parameters. As an example of the application of the hyperfunction ∣x∣^α H(x) we shall show that Hadamard’s finite part of a divergent integral can be introduced in a natural way.