Abstract
Let us begin with the function f(x) defined by \(f(x) = \left\{ {\begin{array}{*{20}{c}} {{{(x - a)}^\alpha }{{(x - b)}^\beta },{\text{ a < x < b}},} \\ {0,{\text{ }}x < a,x > b.} \end{array}} \right.\). If Re α > -1, Re β > -1, then f(x) is absolutely integrable, so f(x) can be reinterpreted as a hyperfunction. If Re α ≤ -1 and/or Re β ≤ -1, f(x) cannot be reinterpreted as a hyperfunction as it stands. What can be done in such as case? We might consider the following method. Notice that f(x) has only two singularities x = a and x = b. So, we choose a point c such that a < c < b and consider the functions
Then,
. The singular points of f 1(x) are x = a and x = c, but x = c is a simple discontinuity. The singular point x = a corresponds to the singularity of xα at x = O. Similarly, the singular points of f 2(x) are the discontinuity at x = c and x = b which corresponds to the singularity of x β at x = O. Therefore, f 1(x) and f 2(x) are simpler than f(x) itself, so that it may be convenient to consider hyperfunctions corresponding to f 1 (x) and f 2(x) and to combine them to obtain the hyperfunction corresponding to f(x).
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© 1992 Springer Science+Business Media Dordrecht
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Imai, I. (1992). Analytic Continuation and Projection of Hyperfunctions. In: Applied Hyperfunction Theory. Mathematics and Its Applications (Japanese Series), vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2548-2_11
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DOI: https://doi.org/10.1007/978-94-011-2548-2_11
Publisher Name: Springer, Dordrecht
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