Distributions Defined by Divergent Integrals

Ram P. Kanwal
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Ram P. Kanwal: The Pennsylvania State University, Department of Mathematics

Chapter Chapter 4 in Generalized Functions Theory and Technique, 1998, pp 71-98 from Springer

Abstract: Abstract In the previous chapters we have defined various singular distributions. One of them is Pf(1/x), defined in Example 4 of Section 2.4. The function 1/x is not integrable on any neighborhood of the origin. We succeeded in regularizing this function by defining the functional Pf(1/x) by the principal value of the singular integral defined by the quantity $$\left\langle \phi ,1/x \right\rangle$$ . The aim of this chapter is to extend this idea and to regularize various singular integrals and thereby define the coresponding distributions. Let us start with a simple example.

Keywords: Integrable Function; Delta Function; Analytic Continuation; Simple Polis; Homogeneous Function (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-1-4684-0035-9_4

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